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Tuning and
Temperament

Historical
Survey

J. Murray Barbour

East Lansing
Michigan State College Press

1951

BY MICHIGAN STATE COLLEGE PRESS
East Lansing

2741.9

usic

LITHOPRINTED IN THE UNITED STATES OF AMERICA BY
CUSHING-MALLOY, INC., ANN ARBOR, MICHIGAN, 1951

PREFACE

This book is based upon my unpublished Cornell dissertation,
Equal Temperament: Its History from Ramis (1482) to Rameau
(1737), Ithaca, 1932. As the title indicates, the emphasis in the
dissertation was upon individual writers. In the present work
the emphasis is on the theories rather than on their promulga-
tors. Since a great many tuning systems are discussed, a sepa-
rate chapter is devoted to each of the principal varieties of tun-
ing, with subsidiary divisions wherever necessary. Even so,
the whole subject is so complex that it seemed best that these
chapters be preceded by a running account (with a minimum of
mathematics) of the entire history of tuning and temperament.
Chapter I also contains the principal account of the Pythagorean
tuning, for it is unnecessary to spend a chapter upon a tuning
system that exists in one form only.

Most technical terms will be defined when they first occur,
as well as in the Glossary, but a few of these terms should be de-
fined immediately. Of small intervals arising from tuning, the
comma is the most familiar. The ordinary (syntonic or Ptole-
maic) comma is the interval between a just major third, with
ratio 5:4, and a Pythagorean ditone or major third, with ratio
81:64. The ratio of the comma (the ratio of an interval is ob-
tained by dividing the ratio of the higher pitch by that of the lower)
is 81:80.

The Pythagorean (ditonic) comma is the interval between six
tones, with ratio 531441:262144, and the pure octave, with ratio
2:1. Thus its ratio is 531441:524288, which is approximately
74:73. The ditonic comma is about 12/11 as large as the syn-
tonic comma. In general, when the word comma is used without
qualification, the syntonic comma is meant.

There is necessarily some elasticity in the manner in which
the different tuning systems are presented in the following chap-
ters. Sometimes a writer has described the construction of a
monochord, a note at a time. That can be set down easily in the
form of ratios. More often he has expressed his monochord as
a series of string-lengths, with a convenient length for the fun-
damental. (Except in the immediate past, the use of vibration
numbers, inversely proportional to the string-lengths, has been
so rare that it can be ignored.) Or he may speak of there being
so many pure fifths, and other fifths flattened by a fractional

PREFACE

part of the comma. Such systems could be transformed into
equivalent string -lengths, but this has not been done in this book
when the original writer had not done so.

Systems with intervals altered by parts of a comma can be
shown without difficulty in terms of Ellis' logarithmic unit called
the cent, the hundredth part of an equally tempered semitone, or
1/1200 part of an- octave. Since the ratio of the octave is 2:1,
the cent is 21/120° . As a matter of fact, such eighteenth century
writers on temperament as Neidhardt and Marpurg had a tuning
unit very similar to the cent: the twelfth part of the ditonic
comma, which they used, is 2 cents, thus making the octave con-
tain 600 parts instead of 1200.

The systems originally expressed in string- lengths or ratios
may be translated into cents also, although with greater difficulty <=
They have been so expressed in the tables of this book, in the be-
lief that the cents representation is the most convenient way of
affording comparisons between systems. In systems where it
was thought they would help to clarify the picture, exponents have
been attached to the names of the notes. With this method, de-
vised by Eitz, all notes joined by pure fifths have the same ex-
ponent. Since the fundamental has a zero exponent, all the notes
of the Pythagorean tuning have zero exponents. The exponent -1
is attached to notes a comma lower than those with zero expo-
nents, i.e., to those forming pure thirds above those in the zero
series. Thus in just intonation the notes forming a major third
would beC°-E-1, etc. Similarly, notes that are pure thirds
lower than notes already in the system have exponents which are
greater by one than those of the higher notes. This use of expo-
nents is especially advantageous in comparing various systems
of just intonation (see Chapter V). It may be used also, with
-fractional exponents, for the different varieties of the meantone
temperament. If the fifth C-G, for example, is tempered by 1/4
comma, these notes would be labeled C° and G" .

A device related to the use of integral exponents lor the notes
in just intonation is the arrangement of such notes to show their

*For a discussion of methods of logarithmic representation of intervals see
J.Murray Barbour, "Musical Logarithms," Scripta Mathematica, VII (1940),
21-31.

PREFACE

harmonic relationships. Here, all notes that are related by fifths,
i.e., that have the same exponent, lie on the same horizontal line,
while their pure major thirds lie in a parallel line above them,
each forming a 45° angle with the related note below. Since the
pure minor thirds below the original notes are lower by a fifth
than the major thirds above them, they will lie in the same higher
line, but will form 135° angles with the original notes. For ex-
ample:

A-1 „F-1 „R-1
C G

This arrangement is especially good for showing extensions of
just intonation with more than twelve notes in the octave, and it
is used for that purpose only in this book (see Chapter VI).

It is desirable to have some method of evaluating the various
tuning systems. Since equal temperament is the ideal system of
twelve notes if modulations are to be made freely to every key,
the semitone of equal temperament, 100 cents, is taken as the
ideal, from which the deviation of each semitone, as C-C , C*-D,
D-E", etc., is calculated in cents. These deviations are then
added and the sum divided by twelve to find the mean deviation
(M.D.) in cents. The standard deviation (S.D.) is found in the
usual manner, by taking the root-mean-square.

It should be added that there may be criteria for excellence
in a tuning system other than its closeness to equal temperament.
For example, if no notes beyond E" or G^ are used in the music
to be performed and if the greatest consonance is desired for the
notes that are used, then probably the 1/5 comma variety of mean-
tone temperament would be the ideal, since its fifths and thirds
are altered equally, the fifths being 1/5 comma flat and its thirds
1/5 comma sharp. If keys beyond two flats or three sharps are
to be touched upon occasionally, but if it is considered desirable
to have the greatest consonance in the key of C and the least in
the key of G", then our Temperament by Regularly Varied Fifths
would be the best. This is a matter that is discussed in detail
at the end of Chapter VTI, but it should be mentioned now.

My interest in temperament dates from the time in Berlin
when Professor Curt Sachs showed me his copy of Mersenne's

vii

PREFACE

Harmonie universelle. I am indebted to Professor Otto Kinkeldey ,
my major professor at Cornell, and to the Misses Barbara Dun-
can and Elizabeth Schmitter of the Sibley Musical Library of the
Eastman School of Music, for assistance rendered during my
work on the dissertation. Most of my more recent research has
been at the Library of Congress. Dr. Harold Spivacke and Mr.
Edward N. Waters of the Music Division there deserve especial
thanks for encouraging me to write this book. I want also to
thank the following men for performing so well the task of read-
ing the manuscript: Professor Charles Warren Fox, Eastman
School of Music; Professor Bonnie M. Stewart, Michigan State
College; Dr. Arnold Small, San Diego Navy Electronics Labora-
tory; and Professor Glen Haydon, University of North Carolina.

J. Murray Barbour

East Lansing, Michigan
November, 1950

Vlll

GLOSSARY

Arithmetical Division — The equal division of the difference be-
tween two quantities, so that the resultant forms an arithme-
tical progression, as 9:8:7:6.

Bonded Clavichord — A clavichord upon which two or more con-
secutive semitones were produced upon a single string.

Cent — The unit of interval measure. The hundredth part of an

1200i

equal semitone, with ratio <y2.

Circle of Fifths — The arrangement of the notes of a closed sys-
tem by fifths, as C, G, D, A, E, etc.

Circulating Temperaments — Temperaments in which all keys
are playable, but in which keys with few sharps or flats are
favored.

Closed System — A regular temperament in which the initial note
is eventually reached again.

Column of Differences — See Tabular Differences.

Comma — A tuning error, such as the interval B^-C in the Py-
thagorean tuning. See Ditonic Comma and Syntonic Comma.

Ditone — A major third, especially one formed by two equal tones,
as L? the Pythagorean tuning (81:64).

Ditonic Comma — The interval between two enharmonically
equivalent notes, as B and C, in the Pythagorean tuning. Its
ratio is 531441:524288 or approximately 74:73, and it is con-
ventionally taken as 24 cents.

Duplication of the Cube — A problem of antiquity, equivalent to
finding two geometrical means between two quantities one of
which is twice as large as the other, or to finding the cube root
of 2.

Exponents — In tuning theory exponents are used to indicate de-
viations from the Pythagorean tuning, the unit being the syn-
tonic comma. Plus values are sharper and minus values flatter
than the corresponding Pythagorean notes. Fractional expo-
nents indicate subdivisions of the comma, as in the meantone
and many irregular temperaments.

GLOSSARY

Equal Temperament — The division of the octave into an equal
number of parts, specifically into 12 semitones, each of which
has the ratio of >f2.

Euclidean Construction — Euclid's method for finding a mean
proportional between two lines, by describing a semicircle
upon the sum of the lines taken as a diameter and then erecting
a perpendicular at the juncture of the two lines.

Fretted Clavichord — See Bonded Clavichord.

Fretted Instruments — Such modern instruments as the guitar
and banjo, or the earlier lute and viol.

Generalized Keyboard — A keyboard arranged conveniently for
the performance of multiple divisions.

Geometrical Division — The proportional division of two quanti-
ties, so that the resultant forms a geometrical progression,
as 27:18:12:8.

Golden System — A system of tuning based on the ratio of the
golden section ( /f5~ - 1):2.

Good Temperaments — See Circulating Temperaments.

Irregular System — Any tuning system with more than one odd-
sized fifth, with the exception of just intonation.

Just— Pure: A term applied to intervals, as the just major third.

Just Intonation — A system of tuning based on the octave (2:1),
the pure fifth (3:2), and the pure major third (5:4).

Linear Correction — The arithmetical division of the error in a

string-length.
Mean-Semitone Temperament — A temperament in which the

diatonic notes are in meantone temperament, and the chromatic

notes are taken as halves of meantones.

Meantone Temperament — Strictly, a system of tuning with flat-
tened fifths (y 5:1) and pure major thirds (5:4). See Varieties
of Meantone Temperament.

GLOSSARY

Meride — Sauveur's tuning unit, 1/43 octave, that is, ^"27 Each
meride was divisible into 7 eptamerides, and each of the ep-
tamerides into 10 decamerides.

Mesolabium — An instrument of the ancients for finding mechan-
ically 2 mean proportionals between 2 given lines. See illus-
tration, p. 51.

Monochord — A string stretched over a wooden base upon which
are indicated the string-lengths for some tuning system; a
diagram containing these lengths; directions for constructing
such a diagram.

Monopipe — A variable open pipe, with indicated lengths for a
scale in a particular tuning system, thus fulfilling a function
similar to that of a monochord.

Multiple Division — The division of the octave into more than 12
parts, equal or unequal.

Negative System — A regular system whose fifth has a ratio
smaller than 3:2.

Positive System — A regular system whose fifth has a ratio
larger than 3:2.

Ptolemaic Comma — See Syntonic Comma.

Pythagorean Comma — See Ditonic Comma.

Pythagorean Tuning — A system of tuning based on the octave
(2:1) and the pure fifth (3:2).

Regular Temperament — A temperament in which all the fifths
save one are of the same size, such as the Pythagorean tuning
or the meantone temperament. (Equal temperament, with all
fifths equal, is also a regular temperament, and so are the
closed systems of multiple division.)

Schisma — The difference between the syntonic and ditonic
commas, with ratio 32805:32768, or approximately 2 cents.

Semi-Meantone Temperament — See Mean-Semitone Tempera-
ment.

GLOSSARY

Sesqui The prefix used to designate a superparticular ratio,

as sesquitertia (4:3).

Sexagesimal Notation — The use of 60 rather than 10 as a base
of numeration, as in the measurement of angles.

Split Keys — Separate keys on a keyboard instrument for such a
pair of notes as G* and Ab.

String- Length — The portion of a string on the monochord that
will produce a desired pitch.

Subsemitonia — See Split Keys.

Superparticular Ratio — A ratio in which the antecedent exceeds
the consequent by 1, as 5:4. See Sesqui-.

Syntonic Comma — The interval between a just major third (5:4)
and a Pythagorean third (81:64). Its ratio is 81:80 and it is
conventionally taken as 22 cents.

Tabular Differences - The differences between the successive
terms in a sequence of numbers, such as a geometrical pro-
gression.

TemPer - To vary the pitch slightly. A tempered fifth is spe-
cifically a flattened fifth.

Temperament — A system, some or all of whose intervals can-
not be expressed in rational numbers.

A Tuning - A system all of whose intervals can be expressed in
rational numbers.

Tuning Pipe — See Monopipe.,

Unequal Temperament — Any temperament other than equal
temperament, particularly the meantone temperament or some
variety thereof.

Varieties of Meantone Temperament — Regular temperaments
formed on the same principle as the meantone temperament,
with flattened fifths and (usually) sharp thirds.

Wolf Fifth - The dissonant fifth, usually G#-Eb (notated as a
diminished sixth), in any unequal temperament, such as the
meantone v/olf fifth of 737 cents.

xii

CONTENTS

Preface Page v

Glossary ix

I. History of Tuning and Temperament 1

II. Greek Tunings 15

III. Meantone Temperament 25

Other Varieties of Meantone Temperament .... 31

IV. Equal Temperament 45

Geometrical and Mechanical Approximations ... 49

Numerical Approximations 55

V. Just Intonation 89

Theory of Just Intonation 102

VI. Multiple Division 107

Equal Divisions 113

Theory of Multiple Division 128

VII. Irregular Systems 133

Modifications of Regular Temperaments 139

Temperaments Largely Pythagorean 151

Divisions of Ditonic Comma 156

Metius' System 177

"Good" Temperaments 178

VIII. From Theory to Practice 185

Tuning of Keyboard Instruments 188

Just Intonation in Choral Music 196

Present Practice 199

Literature Cited 203

Index 219

xiii

CONTENTS

LIST OF ILLUSTRATIONS

Frontispiece: Fludd's Monochord, with Pythagorean
Tuning and Associated Symbolism

Fig. A. Schneegass' Division of the Monochord 38

B. The Mesolabium 51

C. Roberval's Method for Finding Two Geo-

metric Mean Proportionals 52

D. Nicomedes' Method for Finding Two

Geometric Mean Proportionals 53

E. Strahle's Geometrical Approximation for

Equal Temperament 66

F. Gibelius' Tuning Pipe 86

G. Mersenne's Keyboard with Thirty-One

Notes in the Octave 109

H. Ganassi's Method for Placing Frets on

the Lute and Viol 142

I. Bermudo's Method for Placing Frets on

the Vihuela 165

Chapter I. HISTORY OF TUNING AND TEMPERAMENT

The tuning of musical instruments is as ancient as the musical
scale. In fact, it is much older than the scale as we ordinarily
understand it. If primitive man played upon an equally primitive
instrument only two different pitches, these would represent an
interval of some sort — a major, minor, or neutral third; some
variety of fourth or fifth; a pure or impure octave. Perhaps his
concern was not with interval as such, but with the spacing of
soundholes on a flute or oboe, the varied lengths of the strings
on a lyre or harp. Sufficient studies have been made of extant
specimens of the wind instruments of the ancients, and of all
types of instruments used by primitive peoples of today, for
scholars to come forward with interesting hypotheses regarding
scale systems anterior to our own. So far there has been no
general agreement as to whether primitive man arrived at an
instrumental scale by following one or another principle, several
principles simultaneously, or no principle at all. Since this is
the case, there is little to be gained by starting our study prior
to the time of Pythagoras, whose system of tuning has had so
profound an influence upon both the ancient and the modern world.

The Pythagorean system is based upon the octave and the
fifth, the first two intervals of the harmonic series. Using the
ratios of 2:1 for the octave and 3:2 for the fifth, it is possible to
tune all the notes of the diatonic scale in a succession of fifths
and octaves, or, for that matter, all the notes of the chromatic
scale. Thus a simple, but rigid, mathematical principle under-
lies the Pythagorean tuning. As we shall see in the more detailed
account of Greek tunings, the Pythagorean tuning perse was used
only for the diatonic genus, and was modified in the chromatic
and enharmonic genera. In this tuning the major thirds are a
ditonic comma (about 1/9 tone) sharper than the pure thirds of
the harmonic series. When the Pythagorean tuning is extended
to more than twelve notes in the octave, a sharped note, as G#,
is higher than the synonymous flatted note, as A".

The next great figure in tuning history was Aristoxenus, whose
dispute with the disciples of Pythagoras raised a question that is
eternally new: are the cogitations of theorists as important as

TUNING AND TEMPERAMENT

the observations of musicians themselves? His specific conten-
tion was that the judgment of the ear with regard to intervals
was superior to mathematical ratios. And so we find him talking
about "parts'7 of an octave rather than about string-lengths „ One
of Aristoxenus7 scales was composed of equal tones and equal
halves of tones. Therefore Aristoxenus was hailed by sixteenth
century theorists as the inventor of equal temperament. How-
ever, he may have intended this for the Pythagorean tuning, for
most of the other scales he has expressed in this unusual way
correspond closely to the tunings of his contemporaries. From
this we gather that his protest was not against current practice,
but rather against the rigidity of the mathematical theories.

Claudius Ptolemy, the geographer, is the third great figure in
early tuning history. To him we are in debt for an excellent
principle in tuning lore: that tuning is best for which ear and
ratio are in agreement. He has made the assumption here that
it is possible to reach an agreement. The many bitter arguments
between the mathematicians and the plain musicians, even to our
own day, are evidence that this agreement is not easily obtained,
but may actually be the result of compromise on both sides. To
Ptolemy the matter was much simpler. For him a tuning was
correct if itused superparticular ratios, such as 5:4, 11:10, etc.
All of the tuning varieties which he advocated himself are con-
structed exclusively with such ratios. To us, nearly 2000 years
later, his tunings seem as arbitrary as was that of Pythagoras.

Ptolemy's syntonic diatonic has especial importance to the
modern world because it coincides with just intonation, a tuning
system founded on the first five intervals of the harmonic series
— octave, fifth, fourth, major third, minor third. Didymus' dia-
tonic used the same intervals, but in slightly different order. If
it could be shown that Ptolemy favored his syntonic tuning above
any of the others which he has presented, the adherents of just
intonation from the sixteenth century to the twentieth century
would be on more solid ground in hailing him as their patron
saint. Actually he approved the syntonic tuning because its ratios
are superparticular; but so are the ratios of three of the four
other diatonic scales he has given.

Just intonation, in either the Ptolemy or the Didymus ver-

HISTORY OF TUNING AND TEMPERAMENT

sion, was unknown throughout the Middle Ages. Boethius dis-
cussed all three of the above-mentioned authorities on tuning,
but gave in mathematical detail only the system of Pythagoras.
It was satisfactory for the unisonal Gregorian chant, for its small
semitones are excellent for melody and its sharp major thirds
are no drawback. Even when the first crude attempt at harmony
resulted in the parallel fourths and fifths of organum,the Pytha-
gorean tuning easily held its own.

But, later, thirds and sixths were freely used and were con-
sidered imperfect consonances rather than dissonances. It has
been questioned whether these thirds and sixths were as rough as
they would have been in the strict Pythagorean tuning, or whether
a process of softening (tempering) had not already begun. At
least one man, the Englishman Walter Odington, had stated that
consonant thirds had ratios of 5:4 and 6:5, and that singers intui-
tively used these ratios instead of those given by the Pythagorean
monochord. In reply one might note that some theorists continued
to advocate the Pythagorean tuning for centuries after the com-
mon practice had become something quite different. If it was
good enough for them, surrounded as they were by other, less
harsh, tuning methods, it must have sufficed for most of those
who lived in an age when no other definite system of tuning was
known.

The later history of the Pythagorean tuning makes interesting
reading.* It was still strongly advocated in the early sixteenth
century by such men as Gafurius and Ornithoparchus, and formed
the basis for the excellent modification made by Grammateus
and Bermudo. At the end of the century Papius spoke in its favor,
and so, forty years later, did Robert Fludd. In the second half
of the seventeenth century Bishop Caramuel, who has the inven-
tion of "musical logarithms" to his credit, said that "very many"
(plurimi) of his contemporaries still followed in the footsteps of
Pythagoras. Like testimony was given half a century later from
England, where Malcolm wrote that "some and even the Generality
. . . tune not only their Octaves, but also their 5ths as perfectly . . .
Concordant as their Ear can judge, and consequently make their

!See J. Murray Barbour," The Persistence of the Pythagorean Tuning Sys-
tem," Scripta Mathematica, I (1933), 286-304.

TUNING AND TEMPERAMENT

4ths perfect, which indeed makes a great many Errors in the other
Intervals of 3rd and 6th." After another half century we find
Abbe Roussier extolling "triple progression," as he called the
Pythagorean tuning, and praising the Chinese for continuing to
tune by perfect fifths.

Like the systems of Agricola in the sixteenth century and of
Dowland in the early seventeenth century, many of the numerous
irregular systems of the eighteenth century contained more pure
than impure fifths. The instruments of the violin family, tuned
by fifths, have a strong tendency toward the Pythagorean tuning.
And a succession of roots moving by fifths is the basis of our
classic system of harmony from Rameau to Prout and Goetschius.
Truly the Pythagorean tuning system has been long-lived, and is
still hale and hearty!

To return to the fifteenth century and the dissatisfied per-
formers: Almost certainly some men did dislike the too-sharp
major thirds and the too-flat minor thirds so much that they at-
tempted to improve them. But history has preserved no record
of their experiments. And the vast majority must have still been
using the Pythagorean system, with all its imperfections, when
Ramis de Pareja presented his tuning system to the world.

To be sure, Ramis did not present himself as the champion
of a tremendous innovation. He was not a Luther nailing his
ninety-five theses to the church door. His tuning was offered as
a method which would be easier to work out on the monochord,
and thus would be of greater utilitarian value to the singer, than
was the Pythagorean tuning, with its cumbersome ratios. Al-
though Ramis' monochord contained four pure thirds, with ratio
5:4, it was not the usual form of just intonation applied to the
chromatic octave, in which eight thirds will be pure. It is rather
to be considered an irregular tuning, combining features of both
the Pythagorean tuning and just intonation. Some of Ramis' con-
temporaries assailed his tuning method, but his pupil Spataro
explained it as a sort of temperament of the Pythagorean tuning.
From these polemics arose the entirely false notion that Ramis
was an advocate of equal temperament. ^ But he is worthy of our

2lt occurs, for example, in such a general work as Sir James Jeans' Science
and Music (New York, 1937).

HISTORY OF TUNING AND TEMPERAMENT

respect as the first of a long line of innovators and reformers in
the field of tuning.

As the words "tuning" and "temperament" aye used today, the
former is applied to such systems as the Pythagorean and just,
in which all intervals may be expressed as the ratio of two in-
tegers. Thus for any tuning it is possible to obtain a monochord
in which every string-length is an integer. A temperament is a
modification of a tuning, and needs radical numbers to express
the ratios of some or all of its intervals. Therefore, in mono-
chords for temperaments the numbers given for certain (or all)
string-lengths are only approximations, carried out to a partic-
ular degree of accuracy. Actually it is difficult in extreme cases
to distinguish between tunings and temperaments. For example,
Bermudo constructed a monochord in which the tritone G-C#
has the ratio 164025:115921. This differs by only 1/7 per cent
from the tritone of equal temperament, and in practice could not
have been differentiated from it. But his system, which consists
solely of linear divisions, should be called a tuning rather than
a temperament.

It is not definitely known when the practice of temperament
first arose in connection with instruments of fixed pitch, such as
organs and claviers. Even in tuning an organ by Pythagorean
fifths and octaves, the result would not be wholly accurate if the
tuner's method was to obtain unisons between pitches on a mono-
chord and the organ pipes. This would be a sort of unconscious
temperament. More consciously he may have tried to improve
some of the harsh Pythagorean thirds by lopping a bit off one
note or another. Undoubtedly this was being done during the
fifteenth century, for we find Gafurius, at the end of that century,
mentioning that organists assert that fifths undergo a small dimi-
nution called temperament (participata).^

We have no way of knowing what temperament was like in
Gafurius' age; but it is my belief that this diminution which Ga-
furius characterized as "minimae ac latentis incertaeque quo-
demmodo quantitatis" was actually so small that organs so tuned
came closer to being in equal temperament than in just intonation

3Franchinus Gafurius, Practica musica (Milan, 1496), Book 2, Chapter 3.

TUNING AND TEMPERAMENT

or the meantone temperament. This belief is substantiated by
two German methods of organ temperament which appeared in
print a score of jgears later than Gafurius' tome. The earlier of
the two was Arnold Schlick's temperament, an irregular method
for which his directions were somewhat vague, but in which there
were ten flattened and two raised fifths, as well as twelve raised
thirds. Shohe Tanaka's description of Schlick's method4 as the
meantone temperament is wholly false; for in the latter the eight
usable thirds are pure. Actually, from Schlick's own account,
the method lay somewhere between the meantone temperament
and the equal temperament. More definite and certainly very
near to equal temperament was Grammateus' method, in which
the white keys were in the Pythagorean tuning and the black keys
were precisely halfway between the pairs of adjoining white keys.

Just what the players themselves at this time understood by
equal semitones is not known. Perhaps they would have been
satisfied with a tuning like that of Grammateus, with ten semi-
tones equal and the other two smaller. The first precise math-
ematical definition of equal temperament was given by Salinas:
"We judge this one thing must be observed by makers of viols,
so that the placing of the frets may be made regular, namely that
the octave must be divided into twelve parts equally proportional,
which twelve will be the equal semitones. "5 To facilitate con-
structing this temperament on the monochord, Salinas advised
the use of the mesolabium, a mechanical method for finding two
mean proportionals between two given lines . Zarlino also gave
mechanical and geometric methods for finding the mean propor-
tionals, intended primarily for the lute. (Zarlino did include,
however, Ruscelli's enthusiastic plea that all instruments, even
organs, should be tuned equally.) The history of equal tempera-
ment, then, is chiefly the history of its adoption upon keyboard
instruments.

4aStudien im Gebiete der reinen Stimmung," Vierteljahrsschrift fiir Musik-
wissenschaft, VI (1890), 62, 63.

5Francisco Salinas, De musica libri VH (Salamanca, 1577), p. 173.
6

HISTORY OF TUNING AND TEMPERAMENT

Neither Salinas nor Zarlino gave monochord lengths for equal
temperament, although the problem was not extremely difficult:
to obtain the 12th root of 2, take the square root twice and then
the cube root. The first known appearance in print of the correct
figures for equal temperament was in China, where Prince Tsai-
yii's brilliant solution remains an enigma, since the music of China
had no need for any sort of temperament. More significant for
European music, but buried in manuscript for nearly three cen-
turies, was Stevin's solution. As important as this achievement
was his contention that equal temperament was the only logical
system for tuning instruments, including keyboard instruments.
His contemporaries apologetically presented the equal system as
a practical necessity, but Stevin held that its ratios, irrational
though they may be, were "true" and that the simple, rational
values such as 3:2 for the fifth were the approximations! In his
day only a mathematician (and perhaps only a mathematician not
fully cognizant of contemporary musical practice) could have
made such a statement. It is refreshingly modern, agreeing
completely with the views of Schbnberg and other advanced theo-
rists and composers of our day.

The most complete and important discussion of tuning and
temperament occurs in the works of Mersenne. There, in addition
to his valuable contributions to acoustics and his descriptions of
instruments, Mersenne ran the whole gamut of tuning theory. He
expressed equal temperament in numbers, indicated geometrical
and mechanical solutions for it, and finally put it upon the prac-
tical basis of tuning by beats as used today. Fully as catholic is
his list of instrumental groups for which this temperament should
be used: all fretted instruments, all wind instruments, all key-
board instruments, and even percussion instruments (bells)."
The widespread influence of Mersenne's greatest work, Harmonie
universelle (Paris, 1636 - 37), undoubtedly helped greatly to
popularize a tuning that was then still considered as suitable for
lutes and viols only.

The first really practical approximation for equal tempera -

Gjohann Philip Albrecht Fischer, Verhandlung van de Klokken en het Klokke-
Spel (Utrecht, 1738), p. 19, gave a bell temperament, with C equal to 192.000.
This was equal temperament, with a few minor errors.

TUNING AND TEMPERAMENT

ment had been presented by Vincenzo Galilei half a century before
Mersenne. He showed that the ratio of 18:17 was convenient in
fretting the lute. Since references to this size of semitone cover
two and a half centuries, it is probable that it has been used even
longer by makers of lutes, guitars, and the like. Of course the
repeated use of the 18:17 ratio would not give an absolutely pure
octave, but a slight adjustment in the intervals would correct the
error. Galilei's explanation of the reason for equal semitones
on the lute is logical and correct: Since the frets are placed
straight across the six strings, the order of diatonic and chro-
matic semitones is the same on all strings. Hence, in playing
chords, C* might be sounded on one string and DD on another,
and this will be a very false octave unless the instrument is in
equal temperament.

Vicentino had referred to a serious difficulty that arose from
the common practice of having one kind of tuning (meantone) for
keyboard instruments and another (equal) for fretted instruments.
Since the pitches were so divergent, there was dissonance when-
ever the two groups were used together. By 1600, theorists like
Artusi and Bottrigari said that these different groups of instru-
ments were not used simultaneously because of the pitch diffi-
culties. That is why such large instrumental groups were needed
as those employed in the Ballet Comique de la Reine or in Mon-
teverdi's Orfeo — selected groups of like instruments sounded
well, but the mixture of different tunings made tuttis impracti-
cable. It would seem that this consideration would have brought
about the universal adoption of equal temperament long before it
did come. However, after the unfretted violins became the back-
bone of the seventeenth century orchestra, their flexibility of in-
tonation made this problem less pressing than when lutes and
viols had been opposed to organs and claviers.

Before we leave the sixteenth century, we should examine the
contribution to tuning history for which Vicentino is especially
known. His archicembalo was an instrument with six keyboards,
with a total of thirty-one different pitches in the octave. He de-
scribed its tuning as that of the "usage and tuning common to all
the keyboard instruments, as organs, cembali, clavichords, and

HISTORY OF TUNING AND TEMPERAMENT

the like. "^ This would have been the ordinary meantone temper-
ament, in which the fifths were tempered by 1/4 comma. Huy-
ghens, a century and a half after Vicentino, showed that there was
very close correspondence between a system in which the octave
is divided into thirty-one logarithmically equal parts and the
meantone system, similarly extended to thirty- one parts.

A simpler type of multiple division was the cembalo with
nineteen notes in the octave. Both Zarlino and Salinas intended
their variants of the meantone temperament (with fifths tempered
by 2/7 and by 1/3 comma respectively) for such an instrument,
and the latter 's temperament would result in an almost precisely
equal division. Praetorius described such an instrument also,
and it has received favor with some twentieth century writers,
especially Yasser.

The best system of multiple division within the limits of prac-
ticability divides the octave into fifty-three parts. This is lit-
erally a scale of commas, and, as such, was suggested by the
ancient Greek writers on the Pythagorean system. Mersenne
and Kircher in the seventeenth century mentioned the system.
Mercator realized its advantages for measuring intervals. But
especial honor should be paid to the nineteenth century English-
man Bosanquet for devising an harmonium with a "generalized
keyboard" upon which the 53-system could be performed.

Other varieties of equal multiple division will be discussed
in Chapter VI, together with a number of unequal divisions, most
of which are extensions of just intonation. Practical musicians
have rejected all of them, chiefly because they are more difficult
to play, as well as being more expensive, than our ordinary key-
boards.

Just intonation, as has already been mentioned, has had few
devotees since the early seventeenth century. The history of the
meantone temperament makes more interesting reading, since
various theorists in addition to Zarlino and Salinas had conflict-
ing ideas as to the amount by which the fifths should be tempered.
Silbermann's temperament of 1/6 comma for the fifths is the
most significant for us, because he represents the more con-

^Nicola Vicentino, L'antica musica ridotta alia moderna prattica (Rome, 1555),
Book 5, Chapter 6.

TUNING AND TEMPERAMENT

servative practice during the time of Bach and Handel. In his
temperament the thirds are slightly sharp, but the wolves are
almost as ravenous as in the Aron 1/4 comma system.

To some extent the final adoption of equal temperament for
an individual organ or clavier might have meant substituting this
temperament for some type of meantone temperament. We are
told that organs in England were still generally in meantone tem-
perament until the middle of the nineteenth century. England
must have lagged behind the Continent in this respect, and it is
quite possible that the change, when it did come, was radical.

But it is more likely that in most cases the change to equal
temperament was made more smoothly than this. The importance
of unequal systems of twelve notes to the octave has been gen-
erally neglected by the casual historians of tuning, to whom only
the Big Four (Pythagorean, just, meantone, and equal) are of
moment. It is my opinion, however, that the unequal systems
were of the greatest possible significance in bringing about the
supremacy of our present tuning system. Reference has already
been made to the early sixteenth century irregular systems of
Schlick and Grammateus. The former resembled the meantone
temperament; the latter was derived from the Pythagorean tuning.
Bermudo repeated Grammateus' tuning, and his own second
method was basically Pythagorean also. Ramis and Agricola
crossed just intonation with the Pythagorean tuning, with fairly
happy issue. Ganassi and Artusi treated just intonation and the
meantone temperament much as Grammateus and Bermudo had
treated the Pythagorean tuning.

Only a few years later than Grammateus, Aron described for
organs the meantone temperament, mentioned above. In it every
fifth save one was tempered by such an amount (1/4 comma, or
about 1/18 semitone) that four fifths less two octaves would pro-
duce a pure major third. Thus arose the system that, with var-
ious modifications, was to be the strongest opponent of equal
temperament, so far as keyboard instruments were concerned,
for two or three hundred years. In the meantone temperament a
sharped note, as G% is lower in pitch than the equivalent flattened
note, as AD, by the great diesis, which is almost half as large
as a semitone.

HISTORY OF TUNING AND TEMPERAMENT

After Aron's time the meantone temperament, or some similar
system, was generally accepted for organ and clavier. But there
were a few dissenting voices. One was that of his exact contem-
porary Lanfranco, whose practical tuning rules for keyboard in-
struments seem to agree with no system other than equal tem-
perament. Another was that of Fogliano, who was apparently the
first sixteenth century writer to follow Ramis' lead and use in a
tuning system both the pure fifths and the pure thirds of just in-
tonation. But there is a difference; for he realized that the triads
on D and BD would be hopelessly out of tune in such a system,
and therefore recommended that there be a mean D and Bb, each
differing by half a comma from a pair of just pitches. These two
mean pitches hint at Aron's meantone system. Otherwise this is
what we ordinarily understand just intonation to be. Ironically
enough, Fogliano's method, although containing more perfect
thirds than Ramis' did, is far inferior to it if one goes beyond the
ordinary bounds of two flats and three sharps. Beyond these
bounds lay in wait the howling wolves, to muffle whose voices
was the task of many a later worker in this field.

Fogliano had no immediate followers as an advocate of just
intonation, since the following generation was more concerned
with temperament. But almost a century later, certain mathe-
maticians — as Galileo, de Caus,and Kepler — proclaimed again
the validity of pure thirds and fifths. Occasional lone figures,
both mathematicians and music theorists, were to speak in favor
of just intonation, even until our own day. But it is significant
that the great music theorists, such as Zarlino, Mersenne, and
Rameau, presented just intonation as the theoretical basis of the
scale, but temperament as a practical necessity. Equally great
mathematicians with some understanding of music, from Stevin
to Max Planck, have hailed temperament.

From the middle of the sixteenth century, all the theorists
agreed that the fretted instruments, lutes and viols, were tuned
in equal temperament. Vicentino made the first known reference
to this fact, going so far as to state that both types of instrument
had been so tuned from their invention. If we may believe pic-
torial evidence, especially that of the Flemish painters, so me-
ticulous about detail, frets were adjusted to equal temperament

TUNING AND TEMPERAMENT

as early as 1500, although there is not complete agreement on
this point.

In the National Gallery in London, for example, there are
several paintings in which the position of frets is shown plainly.
A Concert, by Ercole de Roberti (1450-96), contains a nine-
stringed lute and a small four-stringed viol, both apparently in
equal temperament. Marco Marziale's Madonna and Child En-
throned with Saints, painted between 1492 and 1507, has an eleven-
stringed lute with intervals equally proportional. And The Am-
bassadors, painted by Hans Holbein the Younger in 1533, has a
six-stringed lute, again in equal temperament. Negative evidence
is furnished by a painting by the early sixteenth century painter
Ambrogio de Predis, whose Angel Playing on a Musical Instru-
ment is playing a nine- stringed lute on which the semitones run
large, small, small, large, and then three equal, as if the notes
might have been C, C*, D, E , E, etc.

Because of the ease of tuning perfect fifths, the Pythagorean
tuning has been the foundation of many of the later irregular sys-
tems, including that of Kirnberger. It also had some importance
for such sophisticated writers as Werckmeister, Neidhardt, and
Marpurg, whose systems with subtly divided commas were di-
rected to the intellect rather than to the ear of the practical mu-
sician.

It becomes apparent, however, from the works of the men just
mentioned that an instrument that was "well tempered" was not
necessarily tempered equally. The title of Bach's famous "48"
meant simply that the clavier was playable in all keys. Werck-
meister and Neidhardt explained clearly that in their systems
the key of C would be the best and D*3 the worst, with the conso-
nance of the other keys somewhere between these extremes.

Mersenne's and Rameau's modification of the 1/4 comma
meantone temperament resembles somewhat the "good" temper-
aments of Werckmeister and Neidhardt, and Gallimard, with the
aid of logarithms, reached a very similar goal. Perhaps the best
of these many irregular systems was Thomas Young's second
method, in which six fifths are perfect, and the other six are tuned

HISTORY OF TUNING AND TEMPERAMENT

flat by 1/6 Pythagorean comma, as in Silbermann's tuning. This
would have been simpler to construct by ear than most of the
systems, and does have an orderly progression from good to poor
tuning as one departs from the most common keys.

In almost all of these irregular systems, from Grammateus
to Young, all the major thirds were sharp to some extent, thus
differing from just intonation and the meantone temperament, in
which the usable thirds were perfect and the others very harsh.
For the practical musician it would have been an easy matter, as
time went on, to tune the "common" thirds still sharper, so that
all the thirds would be equally sharp, and his instrument would
be substantially in equal temperament. Probably this is exactly
what did happen.

The recorded opposition to equal temperament on the part of
such men as Werckmeister and even Sebastian Bach was to the
rigorous mathematical treatment implied by the name "gleich-
schwebend." Theirs was a practical approximation to equality,
and, from the keyboard compositions of Bach, it is evident that
his practice must have been as satisfactory as that of our present-
day tuners, else the great majority of his compositions would
have been unbearable.

Chapter II. GREEK TUNINGS

Greek music theory is highly complex and difficult, with its al-
phabetical notation, the dependence of musical rhythm upon poetic
meter, and all the rest of it. Our confusion is not lessened by
the fact that scholars quarrel about the exact interpretation of
the modal scales and that a pitifully scant remnant of the music
itself is available for study today. Fortunately it is possible to
understand the essentials of Greek tuning theories without enter-
ing into the other and more controversial aspects of Greek mu-
sical science. Moreover, it is advisable that the Greek tuning
lore be presented in some detail in order that the attitude of many
sixteenth and seventeenth century theorists may be clarified.

The foundation of the Greek scale was the tetrachord, a de-
scending series of four notes in the compass of the modern per-
fect fourth. Most typical was the Dorian tetrachord, with two
tones and then a semitone, as A G F E or E D C B. Two or more
tetrachords could be combined by conjunction, as the above tetra-
chords would be with E a common note. Or they might be com-
bined by disjunction, as the above tetrachords would be in reverse
order, with a whole tone between B and A. Tetrachords combined
alternately by conjunction and by disjunction correspond to our
natural heptatonic scale.

The Greeks had three genera— diatonic, chromatic, and en-
harmonic. A diatonic tetrachord contained two tones and a semi-
tone, variously arranged, the Dorian tetrachord having the order
shown above, as A G F E. In the chromatic tetrachord the second
string (as G) was lowered until the two lower intervals in the
tetrachord were equal. Thus A G" F E represents the process
of formation better than the more commonly shown A F* F E.
In the enharmonic tetrachord the second string was lowered still
further until it was in unison with the third string; the third string
was then tuned half way between the second and fourth strings.
In notes the enharmonic tetrachord would be A G"" F E or A F
F E. Thus in the chromatic tetrachord there were the consecu-
tive semitones that we associate with the modern chromatic
genus; but the enharmonic tetrachord contained real quarter
tones, whereas our enharmonically equivalent notes, as F*5 and
E, differ by a comma, 1/9 tone, or at most by a diesis, 1/5 tone.

TUNING AND TEMPERAMENT

Claudius Ptolemy has presented the most complete list of
tunings advocated by various theorists, including himself. *
These (with one exception to be discussed later) were shown by
the ratios of the three consecutive intervals that constituted the
tetrachord, and also by string- lengths for the octave lying be-
tween 120 and 60, using sexagesimal fractions where necessary.
The octave is the Dorian octave, as from E to E, with B-A the
disjunctive tone, always with 9:8 ratio. Ptolemy's tables are
given here (Tables 1-21) with comments following. The frac-
tions have been changed into decimal notation.

Greek Enharmonic Tunings

Table 1. Archytas' Enharmonic

Lengths

60.00 75.00 77.14 80.00 90.00 112.50 115.71 120.00

Names

ECCBA F F E

Ratios

5/4 36/35 28/27 9/8 5/4 36/35 28/27

Cents 1200 814 765 702 498 112 63

Table 2. Aristoxenus' Enharmonic

Lengths 60.00 76.00 78.00 80.00 90.00 114.00 117.00 120.00

b b

Names E C C B A F F E

Parts 16 2 2 10 24 3 3

Cents 1200 791 746 702 498 89 44 0

Table 3. Eratosthenes' Enharmonic

Lengths 60.00 75.00 77.50 80.00 90.00 112.50 116.25 120.00

b b

Names ECCBA F F E

Ratios 5/4 24/23 46/45 9/8 5/4 24/23 46/45

Cents 1200 814 740 702 498 112 38 0

*Claudii Ptolemaei Harmonicorum libri tres. Latin translation by John
Wallis (London, 1699).

GREEK TUNINGS

Greek Chromatic Tunings

Table 4. Archytas' Chromatic

Lengths 60.00 71.11 77.14 80.00 90.00 106.67 115.71 120.00

Names E Db C B A Gb F E

Ratios 32/27 243/224 28/27 9/8 32/27 243/224 28/27

Cents 1200 906 765 702 498 204 63 0

Table 5. Aristoxenus' Chromatic Malakon

Lengths 60.00 74.67 77.33 80.00 90.00 112.00 116.00 120.00

Db C B A Gb F E

23" 23 10 22 4 4

821 761 702 498 119 59 0

Table 6. Aristoxenus' Chromatic Hemiolion

Lengths 60.00 74.00 77.00 80.00 90.00 111.00 115.50 120.00

Names E Db C B A Gb F E

Parts 14 3 3 10 21 42 4*

Cents 1200 837 768 702 498 135 66 0

Table 7. Aristoxenus' Chromatic Tonikon

Lengths 60.00 72.00 76.00 80.00 90.00 108.00 114.00 120.00

Names E Db C B A Gb F E

Parts 12 4 4 10 18 6 6

Cents 1200 884 791 702 498 182 89 0

Names

Parts

14 3

Cents

1200

TUNING AND TEMPERAMENT

Table 8. Eratosthenes' Chromatic

Lengths 60.00 72.00 76.00 80.00 90.00 108.00 114.00 120.00
Names E Db C B A Gb F E

Ratios 6/5 19/18 20/19 9/8 6/5 19/18 20/19

Cents 1200 884 791 702 498 182 89 0

Table 9. Didymus' Chromatic

Lengths 60.00 72.00 75.00 80.00 90.00 108.00 112.50 120.00
Names E Db C B A Gb F E

Ratios 6/5 25/24 16/15 9/8 6/5 25/24 16/15

Cents 1200 884 814 702 498 182 112 0

Table 10. Ptolemy's Chromatic Malakon

Lengths 60.00 72.00 77.14 80.00 90.00 108.00 115.71 120.00
Names E Db C B A Gb F E

Ratios 6/5 15/14 28/27 9/8 6/5 15/14 28/27

Cents 1200 884 765 702 498 182 63 0

Table 11. Ptolemy's Chromatic Syntonon

Lengths 60.00 70.00 76.36 80.00 90.00 105.00 114.55 120.00
Names E Db C B A Gb F E

Ratios 7/6 12/11 22/21 9/8 7/6 12/11 22/21

Cents 1200 933 783 702 498 231 81 0

GREEK TUNINGS

Greek Diatonic Tunings

Table 12. Archytas' Diatonic

Lengths 60.00 67.50 77.14 80.00 90.00 101.25 115.71 120.00
Names EDCBAG F E

Ratios 9/8 8/7 28/27 9/8 9/8 8/7 28/27

Cents 1200 996 765 702 498 294 63 0

Table 13. Aristoxenus' Diatonic Malakon

Lengths 60.00 70.00 76.00 80.00 90.00 105.00 114.00 120.00

Names EDCBAG F E

Parts 10 6 4 10 15 9 6

Cents 1200 933 791 702 498 231 89 0

Table 14. Aristoxenus' Diatonic Syntonon

Lengths 60.00 68.00 76.00 80.00 90.00 102.00 114.00 120.00

Names EDCBAG F E

Parts 8 8 4 10 12 12 6

Cents 1200 983 791 702 498 281 89 0

Table 15. Eratosthenes' Diatonic

Lengths 60.00 67.50 75.94 80.00 90.00 101.25 113.91 120.00

Names EDCBAG F E

Ratios 9/8 9/8 256/243 9/8 9/8 9/8 256/243

Cents 1200 996 792 702 498 294 90 0

TUNING AND TEMPERAMENT

Table 16. Didymus' Diatonic

Lengths 60.00 67.50 75.00 80.00 90.00 101.25 112.50 120.00
Names EDCBAG F E

Ratios 9/8 10/9 16/15 9/8 9/8 10/9 16/15

Cents 1200 996 814 702 498 294 112 0

Table 17. Ptolemy's Diatonic Malakon

Lengths 60.00 68.57 76.19 80.00 90.00 102.86 114.27 120.00
Names EDCBAG F E

Ratios 8/7 10/9 21/20 9/8 8/7 10/9 21/20

Cents 1200 969 787 702 498 265 85 0

Table 18. Ptolemy's Diatonic Toniaion

Lengths 60.00 67.30 77.14 80.00 90.00 101.25 115.71 120.00
Names EDCBA G F E

Ratios 9/8 8/7 28/27 9/8 9/8 8/7 28/27

Cents 1200 996 765 702 498 294 63 0

Table 19. Ptolemy's Diatonic Ditoniaion

Lengths 60.00 67.50 75.94 80.00 90.00 101.25 113.91 120.00

Names EDCBA G F E

Ratios 9/8 9/8 256/243 9/8 9/8 9/8 256/243

Cents 1200 996 792 702 498 294 90 0

Table 20. Ptolemy's Diatonic Syntonon

Lengths 60.00 66.67 75.00 80.00 90.00 100.00 112.50 120.00

Names EDCBAG F E

Ratios 10/9 9/8 16/15 9/8 10/9 9/8 16/15

Cents 1200 1018 814 702 498 316 112 0

GREEK TUNINGS

Table 21. Ptolemy's Diatonic Hemiolon

Lengths 60.00 66.67 73.33 80.00 90.00 100.00 110.00 120.00
Names EDCBA G F E

Ratios 10/9 11/10 12/11 9/8 10/9 11/10 12/11

Cents 1200 1018 853 702 498 316 151 0

Only two of these seventeen or eighteen independent tunings
have had any great influence upon modern music theory— the third
and fourth of Ptolemy's diatonic scales, commonly called the
"ditonic" and the "syntonic." The former is the same as Era-
tosthenes' diatonic, and is the old Pythagorean tuning. It gains
its name from the fact that its major third (ditone) consists of a
pair of equal tones. The latter, the "tightly stretched" in con-
trast to the "soft" (malakon), is what we know as just intonation.
Didymus' diatonic contains the same intervals as Ptolemy's syn-
tonic diatonic, but with the minor tone (10:9) below the major tone
(9:8) instead of the reverse. Didymus' arrangement is the more
logical for constructing amonochord; Ptolemy's in terms of the
harmonic series.

The theorists of the sixteenth and seventeenth centuries,
eager to bolster their ideas with classical prototypes, pointed
out that the just tuning was that of Didymus and Ptolemy. But
they ignored the other diatonic tunings of Ptolemy. They liked
to point out further that in three of the enharmonic tunings the
pure major third (5:4) appears, and in four of the chromatic
tunings the pure minor third (6:5). But only Didymus used en-
harmonic and chromatic tunings that really resembled just into-
nation. His chromatic is tuned precisely as E, C*, C, etc., would
be in just intonation, using the chromatic semitone, 25:24, which
appears in no other tuning. In his enharmonic, not only does the
major third have the ratio 5:4, but the small intervals are "equal"
quarter tones, resulting from an arithmetical division of the
16:15 semitone.* The other nine enharmonic and chromatic tun-
ings depart more or less from Didymus' standard.

TUNING AND TEMPERAMENT

Let us examine more of the peculiarities of these Greek tun-
ings. Archytas has used the same ratio (28:27) for the lowest
interval in each genus, thus having an interval (63 cents) that is
much larger than most of the semitones and smaller than the
quarter tones. The ditonic semitone, 256:243, is about the same
size as Ptolemy's "soft" semitone, 21:20, being a comma smaller
than the syntonic semitone, 16:15. The tones range from mini-
mum, 11:10, through minor, 10:9, and major, 9:8, to maximum,
8:7. Archytas' minor third, 32:27, is a comma larger than the
syntonic third, 6:5, and more than a comma smaller than Ptol-
emy's minor third, 7:6. Eratosthenes' major third, 19:15, is
about the same size as the Pythagorean ditone, 81:64, and is about
a ditonic comma larger than the syntonic third, 5:4.

Ever since his own age a great controversy has raged about
the teachings of Aristoxenus. Instead of using ratios, he divided
the tetrachord into 30 parts, of which, in his diatonic syntonon,
each tone has 12 parts, each semitone 6. Thus, if we are to take
him at his word, Aristoxenus was here describing equal tem-
perament. The sixteenth and seventeenth century theorists were
of the opinion that such was his intention, the advocates of equal
temperament opposing the name of Aristoxenus to that of Ptolemy.

Ptolemy himself did not so understand Aristoxenus' doctrines.
With a fundamental of 120 units, the perfect fourth above has 90
units. Thus, as shown in the tables, Ptolemy subtracted Aris-
toxenus' "parts" from 120. His enharmonic then agrees with that
of Eratosthenes, and his chromatic tonikon with the latter 's
chromatic. But Aristoxenus' diatonic syntonon does not then
quite agree with the Pythagorean (ditonic) diatonic, although the
latter is the only Greek tuning that contains two equal tones.
His diatonic malakon, as Ptolemy has shown it, is unlike any of
the other tunings; whereas in its succession of intervals— large,
medium, small — it resembles Ptolemy's diatonic malakon or
chromatic syntonon.

So it seems quite likely that Aristoxenus did not intend to ex-
press any new tunings by his adding together of parts of a tone,
but simply to indicate in a general way the impression that cur-
rent tunings made upon the ear. But his vagueness has made
possible all sorts of wild speculations. It is even possible, by

GREEK TUNINGS

an improper manipulation of the figures, to argue that Aris-
toxenuswas a proponent of just intonation. Take his enharmonic:
24 + 3 + 3. Add these numbers to 90 in reverse order as before,
getting 90 93 96 120. Then consider these numbers to be fre-
quencies rather than string- lengths. The result is practically
thesameasDidymus': 5/4 x 32/31 x 31/30. Or take Aristoxenus'
diatonic syntonon: 12 + 12 + 6. Treat it as we have just treated
his enharmonic, getting 90 96 108 120. If these are then taken
as frequencies, we have Ptolemy's syntonic, 10/9 x 9/8 x 16/15.

The paramount principle in Ptolemy's tunings was the use of
superparticular proportion, a ratio in which the antecedent ex-
ceeds the consequent by unity. (The Latin prefix "sesqui" is
conveniently used to describe these ratios, e.g., "sesquiquarta,"
meaning 5/4.) Ptolemy used 5/4, 6/5, 7/6, 8/7, etc. Seven of
the eight tunings that bear his own name are constructed entirely
of superparticular proportions, the eighth being the ditonic, or
Pythagorean. Seven tunings that he has ascribed to other writers
also use these ratios exclusively, including all of Didymus' tun-
ings, Archytas' enharmonic and diatonic, and Eratosthenes' chro-
matic (Aristoxenus' chromatic tonikon). In just intonation the
ratios are, of course, superparticular, and this feature only
would have appealed to Ptolemy and his contemporaries. For,
despite the many apparently just intervals used in the given tun-
ings, Ptolemy recognized no consonances other than those of the
Pythagorean tuning— fourth, fifth, octave, eleventh, twelfth, and
fifteenth.

It is easy to obtain, by algebra, all the possible divisions of
the tetrachord built up entirely by superparticular proportions.
(A theory for the superparticular division of tones is shown in
connection with Colonna, in Chapter VIL) Eliminating those in
which one interval is considerably smaller than the smallest
enharmonic quarter tone (46:45), we find that, collectively, the
Greeks had not omitted many possibilities. Other enharmonic
tunings similar to Ptolemy's would be 5/4 x 22/21 x 56/55 and
5/4 x 26/25 x 40/39. Chromatic tunings would include 6/5 x
13/12 x 40/39; 7/6 x 9/8 x 64/63; 7/6 x 10/9 x 36/35; and 7/6 x
15/14 x 16/15. Two others are difficult to classify: 8/7 x 13/12
x 14/13 might best be considered a chromatic tuning, something

TUNING AND TEMPERAMENT

like 14 + 8 + 8 in Aristoxenus' parts. And 8/7 x 8/7 x 49/48 is
undoubtedly a variant of the ditonic tuning, but with a quarter tone
instead of a semitone at the bottom, perhaps 14 + 14 + 2.

In later chapters we shall see many echoes of Greek tuning
methods, not only in such well-known systems as the Pythagorean
and the just, but also in the modified systems, such as Ganassi's,
and in irregular systems, such as Dowland's. Unusual super-
particular intervals are used by Colonna in the poorest tuning
system shown in this book, and also by Awraamoff , whose system
is even worse.

Chapter III. MEANTONE TEMPERAMENT

It is not definitely known when temperament was first used.
Vicentino stated that the fretted instruments had always been
in equal temperament. As for the keyboard instruments, Zar-
lino declared that temperament was as old as the complete
chromatic keyboard. It may well be that some organs in the
fifteenth century had had temperament of a sort, although the
Pythagorean tuning continued to have too many advocates not to
have been dominant in the earlier period. However that may
be, Riemann discovered the first mention of temperament in a
passage from Gafurius' Practica musica (1496). 1 There, among
the eight rules of counterpoint, Gafurius said that organists as-
sert that fifths undergo a small, indefinite amount of diminution
called temperament (participata). Since he was reporting a con-
temporary fact, rather than advocating an innovation, the practice
may have begun decades earlier than his time.

Notice that Gafurius stated that there was nothing regular
about the temperament of his day, nor were the fifths diminished
by any large amount. It seems reasonable to believe that when
organists first became dissatisfied with the extremely sharp
thirds of the Pythagorean tuning, they would go about any altera-
tion of the fifths in a gingerly manner, lopping off a bit here and
a bit there. Grammateus' division of Pythagorean tones into
equal semitones came only twenty-two years after Gafurius'
observation, 2 and ranks very high among irregular systems that
approach equal temperament. It is easy to believe, therefore,
that organs were tuned as well in 1500 as they generally are
today.

Dechales had no authority for stating that Guido of Arezzo
was the father of temperament. «* The association of Ramis^ with

^ugo Riemann, Geschichte der Musiktheorie (Berlin, 1898), p. 327.
2See Chapter VII for Grammateus.

3R. P. Claudius Franciscus Milliet Dechales, Cursus seu mundus mathe-
maticus (Lugduni, 1674), Tomus Tertius, pp. 15-17.

^See Preface and Chapter V.

TUNING AND TEMPERAMENT

temperament is one of the most common misconceptions in the
history of tuning. And, although Schlick's system^ undoubtedly
can properly be described as a temperament, it is just as surely
of an irregular variety. It is well to mention these names, and
discard each of them, before saying that full credit for describ-
ing the meantone temperament must go to Pietro Aron.

InAron's Toscanello^ there is a chapter entitled "Concerning
the temperament (participation) and way of tuning the instrument. "
The tuning is to be made in three successive stages (see Table
22). First, the major third, C-E, is to be made "sonorous and
just." But the fifth C-G is to be made "a little flat." The fifth
G-D is to be similarly flattened, and then A is to be tuned so that
the fifths D-A and A-E are equal. The idea, of course, is to en-
sure an equality of these four fifths, so far as it can be accom-
plished by ear.

Table 22. Aron's Meantone Temperament (1/4 Comma)

7 1.31+1 3 1 2 3+1 5

Names C° C#" D" Eb < E- F * F*" G-* G#" A" Bb * B" C°
Cents 0 76 193 310 386 503 579 697 773 890 1007 1083 1200
M.D. 20.0; S.D. 20.2

In the second stage of tuning, the fifths F-C, B^-F, and
E^-B0 are tempered exactly the same as the diatonic fifths had
been. Finally, in the third stage, C^ and F# are tuned as pure
thirds to A and D respectively. Aron says nothing about G^.
With Kinkeldey we can say that this note "probably belongs to the
third group, "7 and would be tuned as a pure third to E.

The name "meantone" was applied to this temperament be-
cause the tone, as C-D, is precisely half of the pure third, as

5See Chapter VII.

^Toscanello in musica (Venice, 1523); revised edition of 1529 was consulted..

?Otto Kinkeldey, Orgel und Klavier in der Musik des 16. Jahrhunderts (Leip-
zig, 1910), p. 76.

MEANTONE TEMPERAMENT

C-E. Aron said nothing about the division of the comma. But
since the pure E is a syntonic comma lower than the Pythagorean
E,and each fifth is to be tempered by the same amount, the fifths
will all be tempered by 1/4 comma. It is easy to calculate the
ratio of the meantone fifth: the major third has the ratio 5:4;
hence the ratio of the tone will be the square root of this, or
/J5:2. The ratio of the major ninth will be twice the ratio of the
tone, or ^5:1. The ratio of the fifth will be the square root of
the ratio of the ninth, or 4V57l. If we consider the syntonic comma
to be 21.5 cents, a fifth diminished by 1/4 comma will be 702.0-
5.4 = 696.6 cents.

The deviation for the meantone temperament is nearly as
large as for just intonation. That would seem to indicate that
temperament makes for little improvement. Strangely enough,
this is absolutely true, so far as the remote keys are concerned.
However, if the deviation were to be measured only from E^ to
G*, without allowing for the enharmonic uses of notes, the mean-
tone temperament would be an easy victor over just intonation.
That is, if we were computing the deviation of eleven fifths only,
omitting the wolf fifth of 737 cents, the standard deviation for the
meantone temperament would be much smaller than that for just
intonation. But, since our ideal is equal temperament, the de-
viation as computed shows accurately enough how very unsatis-
factory this tuning is when its narrow bounds are overstepped.

The meantone temperament was used from the beginning upon
keyboard instruments only. It was the temperament that Vicen-
tino intended for his Archicembalo when he said that it may be
tuned "justly with the temperament of the flattened fifth, accord-
ing to the usage and tuning common to all the keyboard instru-
ments, as organs, cembali, clavichords, and the like."** Zarlino
called the meantone temperament a "new temperament" and said
that it is "very pleasing for all purposes" when used on key-
board instruments. 9 To divide the major third into two mean
tones, Zarlino advocated the Euclidean construction for a mean

8See Chapter VI.

^Gioseffo Zarlino, Dimostrationi armoniche (Venice, 1571), p. 267.

TUNING AND TEMPERAMENT

proportional, and of course the fifth could be constructed from
the major ninth by the same means.

Verheijen's reply to Stevin's discussion of equal temperament
explained the meantone temperament in detail 10 He even in-
cluded amonochord for it (Table 23), and thus has the distinction
of being the first person, so far as we know, to put its ratios into
figures (cents values as in Aron, Table 22, beginning with F as
503).

Table 23. Verheijen's Monochord for Meantone Temperament
Lengths 10000 9750 8944 8560 8000 7477 7155 6687 6400 5961

7 19 13 1 3

Names F° F#" G" G#" A*"1 Bb+4 b"*~ C~* C#"2 D"
5590 5350 5000

1 5

Eb+- E" Fo

In Spain, Sancta Maria described a practical tuning system
that may have been the same as the meantone tuning. H He said
that on the clavichord and thevihuela (the Spanish lute) each fifth
is to be "a little flat." In fact, the diminution is to be "so small
that it can scarcely be noticed." Since he did not say whether the
thirds were to be pure or a little sharp, we cannot know whether
his system was the real meantone or came nearer equal tem-
perament. However, he held that a tone cannot be divided into
two equal semitones, and consistently made the diatonic semi-
tone larger than the chromatic semitone, as it would be in just
intonation or the meantone temperament.

The first German writer to describe the meantone tempera-
ment was more explicit. This was Michael Praetor ius,* 2 m a

lOsimon Stevin, Van de Spiegeling der Singconst, ed. D. Bierens de Haan
(Amsterdam, 1884). Verheijen's letter is in Appendice A. Both discussion
and reply remained in manuscript for almost three hundred years.

^Tom^s de Sancta Maria, Arte de taner fantasia (Valladolid, 1565), Chapter
53.

12syntagma musicum (Wolfenbuttel, 1618), Vol. II; new edition, 1884-94, pub-
lished as Publikation alterer praktischen und theoretischen Musikwerke,
Band 13, pp. 178 ff.

MEANTONE TEMPERAMENT

chapter on the tuning of the "Regal, Clavicymbel, Symphonien und
dergleichen Instrument." His was a practical system, with major
thirds and octaves pure, and fifths flat. Praetor ius explained
carefully how various intervals are altered by fractional parts
of the comma.

Otto Gibelius^ showed a method for obtaining an approxi-
mately correct monochord for the meantone temperament. First
he made a table in which were shown pairs of numbers differing
by the syntonic comma for every note in a 14-note octave, ex-
tending from A*3 toD#. Then he made an arithmetical division
of each comma, with 3/4, 1/2, or 1/4 comma subtracted from
the larger number, to obtain the tempered value. C, E, G^, and
A*3 needed no temperament (see Table 24). His results check
closely with numbers obtained by taking roots. ^ For example,
his D is 193200; it should be 193196. His G is 144450 instead of
144447. Since the comma is small relative to the intervals of
the scale and since as much as a quarter or a half of it is used,
the error could not be great. An arithmetical division of the
ditonic comma into twelfths in the construction of equal temper-
ament would create greater errors than this for certain notes of
the division.

Table 24. Gibelius' Monochord for Meantone Temperament

Lengths 216000 206720 193200 184896 180562.5 172800 161500 154560
Names C° C#" D" D*"* Eb+* E_1 F+4 F#"

144450 138240 135000 129200 120750 115560 108000
G"5 G#"2 Ab+1 A" Bb+2 B" C°

Lemme Rossi,* ^ writing in the same year as Gibelius, would
have approved the latter 's approximation for the meantone tem-

1,jPropositiones mathematico-musicae (Mlinden, 1666), copperplate opposite
page 14

l^Wolffgang Caspar Printz, Phrynis Mytilenaeus oder der satyrische Com-
ponist (Dresden and Leipzig, 1696), p. 73.

l^sistema musico (Perugia, 1666), p. 59.

TUNING AND TEMPERAMENT

perament, for he himself said that the arithmetical division of
the comma differs "insensibly" from a geometrical division. In
the example that he gave, the geometrical mean between the two
numbers, 31104 and 30720, in the ratio of 81 to 80, is 30911, and
the arithmetical mean is 30912, certainly a negligible difference.
But, he said, the correct string- lengths for the meantone tem-
perament can be obtained both "easily and quickly with the table
of logarithms."

Our final monochord for the meantone temperament proper
will be Rossi's "Numeri del sistema participate "16 He has
given it for a 19-note octave commencing on A (see Table 25).
Since C itself is a tempered value here, we have transposed the
system up a minor third from A to C, selecting those notes that
would belong to the ordinary meantone scale. The number used
for his fundamental had been previously used in a table of just
intonation.

Table 25. Rossi's Monochord for Meantone Temperament

Lengths
Names

41472
C°

24806

39690

23184

37095

22187

34668
Eb+^

20736
C°

33178

-i
E

31008
F+i

29676

F#~2

27734
i

G" .

26542

G#"2

Another sort of approximation connected with the meantone
temperament was given by Claas Douwes.l^ In describing the
bonded clavichord he gave simple ratios (most of them super-
particular) for various intervals that would occur on the same
string. For example, the highest string has C, B, B*3, and A.
C-A is 6:5; B-A, 19:17; Bb-A, 15:14. On the next string, G#-F
is 7:6. Two octaves lower, the ninth string has only two notes,
G# and G, with the ratio 24:23.

Dou\ ^s had explained that his was a tempered system. His
rational ratios are good approximations to the surds of the mean-

16lbid., p. 83.

l^Grondig Ondersoek van de Toonen der Musijk (Franeker, 1699), pp. 98-104.

MEANTONE TEMPERAMENT

tone temperament. His minor third, with ratio 6:5, is 316 cents;
the meantone minor third is 310. His augmented second, 7:6, is
267 cents; the meantone augmented second is 270. His tone,
19:17, is almost 193 cents; the meantone tone is practically the
same. His diatonic semitone, 15:14, is 119 cents; the meantone
diatonic semitone, 117. His chromatic semitone, 24:23, is 74
cents; the meantone chromatic semitone, 76. His system agrees
with itself as well as with the ordinary meantone system. For
example, the tone should be the sum of the diatonic and the chro-
matic semitones, or 15/14 x 24/23. This product is 3420:3059;
his ratio for the tone, 19:17, equals 3420:3060, a close corre-
spondence.

In tracing the later history of the meantone temperament, it
would be easy to name theorists in all the principal European
countries who continued to favor an unequal tuning of keyboard
instruments later than the first quarter of the eighteenth century.
But, unless, like Galin in 1818, they specifically say that they
favor the tuning in which the fifths are tempered by 1/4 syntonic
comma or its equivalent (31-division)/^ we have no right to call
their methods the meantone temperament. This is the fallacy of
so much that has been written on this subject.

Other Varieties of Meantone Temperament

Strictly, there is only one meantone temperament. But theo-
rists have been inclined to lump together under that head all
sorts of systems intended for keyboard instruments. For ex-
ample, the statement often appears in print that in England the
meantone temperament was used for organs until the middle of
the nineteenth century. William Crotch, 1* writing early inthat
century, wrote: "As organs are at present tuned, (with unequal
temperament), keys which have many flats or sharps will not
have a good effect, especially if the time be slow." That state-
ment is enough to cause a host of later English writers to say

18pierre Galin, Exposition d'une nouvelle me'thode pour l'enseignement de la
musique (3rd edition, Bordeaux and Paris, 1862; 1st edition, 1818).

^Elements of Musical Composition (London, 1812), p. 112.

TUNING AND TEMPERAMENT

that Crotch reported the meantone temperament to be in use in
his age.

But later in his book Crotch had this to say: "Unequal tem-
perament is that wherein some of the fifths, and consequently
some of the thirds, are made more perfect than on the equal
temperament, which necessarily renders others less perfect.
Of this there are many systems, which the student is now capable
of examining for himself. "20 jn other words, Crotch is saying
that there was a great diversity in the tuning of organs in his day.

In Chapter VII, "Irregular Systems," twenty-odd men are
mentioned who collectively have described fifty of the "many
systems," none of which is the meantone temperament. In the
present chapter we propose to describe still other systems of
temperament, systems formed on the same general pattern as
meantone temperament. Bosanquet called "regular" a tempera-
ment constructed with one size of fifth. **■ The Pythagorean tun-
ing, equal temperament, meantone temperament— all are regular
systems. The systems that follow are also regular, with values
for the fifth smaller than that of equal temperament and (usually)
larger than that of the meantone temperament. Since their con-
struction is similar, it is easy to describe them as varieties of
the meantone temperament. In all of them, the tone is precisely
half of the major third. No harm will be done by such a nomen-
clature if we realize that these are regular temperaments which
the earlier theorists themselves considered of the same type as
the 1/4 -comma temperament and some of which they preferred
to it.

The first regular temperament to be advocated after the de-
scription of the ordinary meantone temperament was that de-
scribed by Zarlino in which "each fifth remains diminished and
imperfect by 2/7 comma. "22 Although Zarlino showed a mono-
chord with this tuning for the diatonic genus only, he intended it

20lbid., p. 135.

21R.H M. Bosanquet, An Elementary Treatise on Musical Intervals and Tem-
perament (London, 1876), Chapter VTJI.

22cioseffo Zarlino, Istitutioni armoniche (Venice, 1558), pp. 126 ff.

MEANTONE TEMPERAMENT

also for the chromatic genus— by which he meant the ordinary
black keys. He also described an enharmonic genus, having 19
notes to the octave, as applied to a cembalo which Master Dom-
enico Pesarese had made for him. This must have had the same
tuning, although Zarlino did not clearly say so. Most of these
varieties of the meantone temperament will have a smaller de-
viation when applied to a keyboard with 19 or more notes to the
octave than upon the usual keyboard. Zarlino's temperament
corresponds to the 50-division, and, as such, will be discussed
in the chapter on multiple division.

In Table 26, we see the 2/7-comma temperament applied to
a keyboard with 12 notes to the octave. Since the amount of tem-
pering is greater than 1/4 comma, the deviation is greater than
for Aron's system. It is, in fact, a very poor system, and Zar-
lino later admitted it to be inferior to the 1/4- comma system.
The only just interval in it is the chromatic semitone. Tanaka
liked it "because all the imperfect consonances are impure
alike, "23 that is, the major and minor thirds are 1/7 comma flat
(3 cents), and the major and minor sixths are sharp by the same
amount. To construct it on a monochord, Zarlino would use the
questionable virtues of the mesolabium.24

Table 26. Zarlino's 2/7 - Comma Temperament

4 6 82 _12 2 _16 6 +4 10

Names C° C# D" Eb 7 E~7 F 7 F#" ' G~7 G#" 7 A~7 Bb * B- 7 C°
Cents 0 70 191 313 383 504 574 696 817 887 1008 1078 1200
M.D. 25.0; S.D. 25.3

The next variety of meantone temperament is also highly un-
satisfactory when applied to an octave of twelve semitones. This
is the 1/3-comma temperament, the invention of Francisco Sali-
nas, which he described as follows: "The first of them [the other
two were the 2/7-comma and the 1/4-comma temperaments] has
the comma divided into three parts equally proportional, of which

23Shoh^ Tanaka, "Studien im Gebiete der reinen Stimmung," Vierteljahrs-
schrift fur Musikwissenschaft, VI (1890), 65.

24 For an account of the mesolabium, see the second part of Chapter IV.

TUNING AND TEMPERAMENT

the minor tone is increased by one part and the major tone is
decreased by two parts. "25 Salinas showed that his method re-
sults in pure minor thirds, tritone, and major sixth. But the
fifth is diminished by 1/3 comma, and so is the major third. On
the whole this tuning does not compare favorably with the others,
but Salinas added: "Although this imperfection is seen to be
greater than that which is found in the other two temperaments,
nevertheless it is endurable."

Salinas intended his temperament for an octave containing 19
notes, divided into the three genera— diatonic, chromatic, and
enharmonic. His special reason for advocating this tuning was
the ease of realizing it upon the monochord. Seven of the notes
can be obtained by a series of just minor thirds below and above
the fundamental. Thus we obtain C, D#, E°, F#, Gb, A, and B#,
and Salinas has given their string- lengths for the octave 22500
to 11250.

To find the notes D and E, two mean proportionals must be
inserted in the tritone, C— F^. This "will be very easy to those
who know the use of a certain instrument invented by Archimedes,
which is called mesolabium, from finding mean lines by it." The
remainder of the notes can then be obtained by minor thirds
from D and E.

We agree with Salinas that the thirds and especially the fifths
of the 1/3-comma temperament are less pleasing than those of
the other two. But, in addition to its being capable of quicker
tuning than the Zarlinian 2/7-comma method, it has an advantage
possessed by neither of the other methods: it is practically a
closed or cyclic system. Among its 19 notes there is no fifth
containing a wolf; nor are there any discordant thirds. It is an
equal temperament of 19 notes.

In recent times the 19-division has had eloquent advocates,
to whom reference is made in the chapter on multiple division.
Let us see how well the 1/3-comma system is adapted to a 12-
note keyboard. As Table 27 shows, this is the poorest tuning of
all— like Zarlino's method, it is worse than just intonation. How-
ever, too many theorists who have described these two systems
have neglected to add that they are excellent foral9-note octave.

2&De musica libri VII, p. 143.

MEANTONE TEMPERAMENT

Table 27. Salinas' 1 /3 - Comma Temperament

7 2 +1 44-L 2 l 8 +2 5

Names C° C^" D~3 Eb E" F s f#~ G" G#" a" Bb 3 B" C°
Cents 0 64 190 316 379 505 569 695 758 884 1010 1074 1200

M.D. 30.3; S.D. 30.7

It would help us in portraying an orderly development of the
12-note temperaments if we could show that little by little the
temperament of the fifth was reduced from the 1/4 comma of the
meantone temperament to the 1/11 comma (1/12 ditonic comma
equals 1/11 syntonic comma) of equal temperament. Probably
there was such a tendency. But it is only a fortunate accident
that Verheijen included the ratio of the fifth for the 1/5-comma
temperament, together with the ratios for the three temperaments
discussed by Zarlino and Salinas. 26 Verheijen's first ratio for
the fifth is the cube root of 10:3 (1/3-comma temperament); then
the fourth root of 5:1 (1/4-comma); the fifth root of 15:2 (1/5-
comma); the seventh root of 50:3 (2/7-comma). Verheijen's
casual reference to the 1/5-comma temperament indicates that
even then some people were using it. Rossi, a couple of genera-
tions later, also referred briefly to the 1/5-comma temperament,
including it as one of the regular types then in use. ^

The temperament shown in Table 28 has in its favor, like the
1/3-comma temperament, the equal distortion of the fifths and
the major thirds, the former being 1/5 comma flat, the latter
sharp by the same amount. In it the diatonic semitone is pure.

Table 28. 1/5 - Comma Temperament (Verheijen, Rossi)

7 2 3 4 1 6 1 8 3,2

Names C° C#'z D"5 Eb ' e" F ■ P*" G" G#" A" Bb » B_1 C°
Cents 0 83 195 307 390 502 586 698 781 893 1005 1088 1200
M.D. 14.0; S.D. 14.2

''"Simon Stevin, Van de Spiegeling der Singconst, Appendice D.
2'Sistema musico, p. 58.

TUNING AND TEMPERAMENT

The deviation of this temperament is only about two-thirds that
of the 1/4-comma system.

There is an odd reference to the 1/5- comma temperament.
Dechales^S gave a monochord which he called the "Diatonic
scale of Guido of Arezzo." It is, however, a chromatic scale,
and, so far as can be ascertained, has nothing in common with
any of the ideas expressed by Guido.

It seems evident that Dechaleshas intended the monochord in
Table 29 for the 1/5-comma temperament. Its ninth note differs
greatly from the cents value given in the previous table; but the
note is A^ in Dechales' monochord and would naturally be more
than a comma higher than the G^ more commonly used. Other
divergences can be explained by the fact that Dechales has not
expressed his numbers with great accuracy. However, the mean
value for his diatonic semitone is 111.4, against 112.0 for the
1/5-comma temperament; for his chromatic semitone, 84.0 cents
against 83.2. How he reached the conclusion that Guido favored
such a temperament remains a mystery. Actually Dechales him-
self ascribed the 1/4-comma temperament to Guido (rather than
the 1/5-comma), contrary to the evidence of this monochord.

Table 29. Dechales' "Guidonian" Temperament (1/5 - Comma)

Lengths 60 57i 53| 50^ 47| 44^ 42| 40| 37| 35^ 33i 31§ 30

Names C C# D Eb E F F# G Ab A Bb B C

Cents 0 85 194 312 395 502 587 696 808 893 1009 1090 1200

M.D. 13.3; S.D. 13.8

The 1/5-comma variety of meantone temperament comes
close to the 43-division. As such, it is discussed briefly in
Chapter VI, with the principal reference to Sauveur.

Another temperament discussed by Rossi^ has its fifths
flattened by 2/9 comma (see Table 30). He merely called it
"another tempered system," without ascribing it to any theorist.
Romieu identified this temperament with the 31 -division, and thus

28cursus seu mundus mathematicus, p. 20.

29sistema musico, p. 64.

MEANTONE TEMPERAMENT

Table 30. Rossi's 2/9 - Comma Temperament

14 4 282 42 16 2 4 _10

Names C° C#"~ D" E°+S e" F+a F#" G" G#~ 9 A" Bb+9 B_ 9 C°
Cents 0 79 194 308 389 503 582 697 777 892 1006 1085 1200
M.D. 17.0; S.D. 17.2

credited it to Huyghens.^O Actually, as we have already said,
the 1/4-comma temperament comes closest to the 31-division.
But perhaps other writers before Romieu confused these tem-
peraments. For example, Printz^l spoke of a "still earlier"
temperament that takes 2/9 comma from each fifth— earlier,
perhaps, than Zarlino's 2/7- comma temperament, which he had
been previously discussing. He also might have meant Vicentino's
31-division, since there are no early references to the 2/9-
comma temperament.

Since 2/9 is the harmonic mean between 1/4 and 1/5, the de-
viation for this temperament is approximately the mean of the
deviations of the other two temperaments. Like Zarlino's 2/7-
eomma temperament, its third is altered half as much as its
fifth, being 1/9 comma sharp. Its augmented second, as F-G#,
is pure. The 74-division corresponds to the 2/9-comma tem-
perament, and Drobisch liked this division best of all systems
that form their major thirds regularly.

Schneegass gave an interesting geometrical construction for
what was much like the common meantone temperament, but
more like the 2/9-comma temperament. His contention was
that the diatonic semitone contains 3 1/4 "commas" and the
chromatic semitone 2 1/4. (These commas of 35.3 cents have
nothing in common with either the ditonic [23. 5J or the syntonic
[2I.5] comma). Thus the tone contains 5 1/2 commas, and the
octave 5x5 1/2 + 2x3 1/4 = 34 commas. As is shown in Chapter
VI, the 34 -division has fifths that are almost 4 cents too large

30jean-Baptiste Romieu, "Memoire theorique & pratique sur les systemes
tempe'res de musique," Me'moires de Tacademie royale des sciences, 1758,
p. 837.

31Phrynis Mytilenaeus oder der satyrische Componist, p. 88.

32Cyriac Schneegass, Nova & exquisita monochordi dimensio (Erfurt, 1590),
Chapter HI.

TUNING AND TEMPERAMENT

and thirds that are 2 cents too large. But this was not what
Schneegass had in mind. His theoretical fifth had the ratio
160:107, or 696.6 cents, which is precisely the size of the mean-
tone fifth, and he directed that this ratio be used twice to form
the tone.

Then came the application of the doctrine about commas: A
right triangle was to be constructed, with the space of the tone,
G-A, as base, and thrice this length for the altitude (see Fig-
ure A). Note that "space" here does not refer to the total length
of a line, but rather to the distance from one point of division to
another Since 3 1/4:2 1/4 = 13:9, the acute angle at thetopwas
to be divided in the ratio of 13:9, with the larger angle toward
A. The point where this line cut the base was to be G#. Now
tan"1 1/3 = 18° 26', and 13/22 of this angle is 10° 53'. The space
between G# and A, then, would be 3 tan 10° 53' = .57681 of the
space between G and A. From the figures in his table, the divi-
sion was made with extreme care. The ratio in the table of the
space from G* to A to the space from G to A is 15/26 or .57692.
By a series of lines parallel to the base, he cleverly divided the
other tones (Bb-C, C-D, Eb-F, and F-G) into chromatic and
diatonic semitones proportional to the division of G-A.

Fig. A. Schneegass' Division of the Monochord

Reproduced by courtesy of the Sibley Library* of the
Eastman School of Music

MEANTONE TEMPERAMENT

To examine the assumption that Schneegass made, let us des-
ignate as a the angle 10° 53' and as ft the angle 18° 26 ', and as
L the length for the note A. Then the length for G was L + tan a ,
and for G it was L + tan ft . His assumption:

log/L + tanff ): log/ L + tan a \= ft : a

In general this would be only a rough approximation. In this case,
where ft : a = 22:13, it works very well indeed.

Schneegass' actual fifth, G-D,of 698.1 cents is a little larger
than his theoretical fifth of 696.6, and the mean of all 11 good
fifths is 697.2 cents. This last figure is precisely the fifth of
the 2/9 -comma temperament. The mean value of his tones is
194.0 cents, as compared with 194.4 cents of the 2/9- comma
temperament, and his geometrical division of the tones yields
semitones of 113.9 and 80.1 cents, compared with 114.0 and 80.4
cents.

Schneegass' actual fifth has approximately the ratio 226:151,
instead of his theoretical 160:107. It is idle to speculate why his
figures fail to correspond with his theory, or why they agree so
beautifully with the 2/9 -comma temperament. The significant
thing is that they agree so well with themselves, which is an in-
dication of the soundness of his mathematics! There is, how-
ever, one puzzling clue to his division of the tone. Suppose the
space of the tone G-A had been divided arithmetically in the ratio
of 13:9, instead of the more complicated division of the angle
actually used. Then Schneegass' G* would have been at 86.100
instead of at 85.967. This would have made the G* 3.3 cents
lower than in the table, and his tone would have been divided into
semitones of 117.7 and 76.0 cents. Nowthe semitones of the 1/4-
comma temperament are of 117.1 and 76.0 cents respectively.
Thus an arithmetical division of his tones would have come close
to the temperament which is suggested by his theoretical fifth.
However, his actual division (Table 31) with a 15:11 ratio, is
very consistent with itself, as well as with the 2/9 -comma tem-
perament.

TUNING AND TEMPERAMENT

Table 31. Schneegass' Variety of Meantone Temperament

Lengths 90.000 85.967 80.467 75.267 71.867 67.267 64.200 60.133
Names G G# A Bb B C C# D

Cents 0 79 194 309 389 504 585 698

56.300 53.750 50.367 48.083 45.000
Eb E F F* G

812 892 1005 1085 1200

M.D. 16.7; S.D. 16.9

Robert Smith"^ is responsible for three wholly unsatisfactory
varieties of the meantone temperament. He told first of a Mr.
Harrison, who tuned his viol by "taking the interval of the major
third to that of the octave, as the diameter of a circle to its cir-
cumference— It follows from Mr. Harrison's assumption, that
his 3rd major is tempered flat by a full fifth of a comma." If the
ratio of the major third to the octave is l:tr , the third will have
382.0 cents, or be 1/5 comma flat, as Smith said. The fifth will
then be tempered by 3/10 comma. Romieu^ barely mentioned
3/10- and 3/11 -comma temperaments, but did not discuss them
on the ground that they were too like temperaments with unity in
the numerator. Except for a few references to Smith and this
tuning by rr, the 3/10- comma temperament has escaped further
notice (see Table 32).

Table 32. Harrison's 3/10 - Comma Temperament

21_ 3 _9_ 63 2 3_ 12 9 3 3

Names C° C#~10 D" Eb+l° E" F w F*" G"10 G#~~ A~a BD * B" C°
Cents 0 69 191 314 382 504 573 696 764 887 1009 1078 1200
M.D. 26.2; S.D. 26.6

Since 3/10 is about the same as 2/7, the deviation for this
temperament is approximately the same as for Zarlino's, both

■^Harmonics, or the Philosophy of Musical Sounds (Cambridge, 1749), pp. xi,
xii.

3^In Memoires de l'acade'mie royale des sciences, 1758, p. 827.
40

MEANTONE TEMPERAMENT

being inferior to just intonation. It has no special features to
recommend it, since its one natural feature, the Tr ratio, is
something to be determined by ear or by logarithms, and would
not make the construction of a monochord any simpler.

After referring to Harrison's system, as quoted above, Smith
continued, "My third determined by theory, upon the principle of
making all the concords within the extent of every three octaves
as equally harmonious as possible, is tempered flat by one ninth
of a comma; or almost one eighth, when no more concords are
taken into the calculation than what are contained within one oc-
tave." Later he showed that "to have all the concords in four
octaves made equally harmonious," the thirds will be 1/10 comma
flat. 35

With the third flat by 1/9 comma, the fifth will be tempered
by 5/18 comma, a quantity impossible to judge by ear. In the
second temperament, with the third 1/10 comma flat, the fifth
will be 11/40 comma flat. The difference between these values
of the fifth is only 1/360 comma! Therefore the temperaments
would not vary for any note by as much as one cent. For this
reason only the first of Smith's temperaments is shown in Table
33.

Table 33. Smith's 5/18 - Comma Temperament

35 5 5 _10 5 5 5 20 5 +5 25

Names C° C#~18 D" Eb * E" 9 F '» F*" G"15 G#" 9 A" Bb s B"18 C°
Cents 0 72 192 312 384 504 576 696 768 888 1008 1080 1200
M.D. 23.3; S.D. 23.7

Since 5/18 is also approximately the same as 2/7, Smith's
temperament is only a little better than Zarlino's. We have pre-
viously indicated that the 50-division has usually been considered
the equivalent of the 2/7-comma temperament. Smith asserts,
however, that his temperament corresponds to the 50-division,
the error of the fifth in the latter being 41/148 comma. He is
entirely correct in his claim.

Smith did not suggest, however, that the octave be divided into
fifty parts— merely that "a system of rational intervals deduced

35smith, Harmonics, p. 171.

TUNING AND TEMPERAMENT

from dividing the octave into 50 equal parts, ...will differ insensi-
bly from the system of equal harmony." His desire is more
modest— to have at least 21 different pitches in the octave, pro-
perly to differentiate the sharps, naturals, and flats. On the or-
gan and harpsichord this could be done by adding extra pipes and
strings. Performance would be facilitated by having "seven
couples of secondary notes," governed by stops, so that the ap-
propriate notes for a particular piece could be chosen. Of course,
upon an instrument with 19 notes to the octave (the other two
would be of little use), Smith's temperament, like Zarlino's and
Salinas', would be far more acceptable than on the ordinary key-
board. Smith himself considered that ordinary equal tempera-
ment "far exceeds" both the 31- and 50-divisions, because of the
cumbersomeness of the latter systems.

The only other important variety of the meantone temperament
was that practiced by Silbermann and his contemporaries. Ac-
cording to Sorge, Silbermann tempered his fifths by 1/6 comma.36
Since Sorge himself made no distinction between the syntonic
and ditonic commas, we might divide either. If we divide the
ditonic comma, the deviation is precisely the same as for the
Pythagorean tuning, M.D. 11.7, S.D. 11.8. But, for better com-
parison with the other varieties of meantone temperament, let
us divide the syntonic comma. Then the major third is 1/3
comma sharp, and the tritone is pure (see Table 34).

Table 34. Silbermann's 1/6 - Comma Temperament

7 1 ,1 2,1 1 4 1 +1 5

Names C° C#" D" Eb a E" F « F*~ G~« G#"^ A" Bd ' B" C°
Cents 0 89 197 305 394 502 590 698 787 895 1003 1092 1200
M.D. 9.3; S.D. 9.5

RomieuS? adopted the 1/6-comma temperament as his "tem-
perament anacratique," showing its correspondence to the 55-
division. A generation after Romieu, Barca called thistempera-

36Georg Andreas Sorge, Gesprach zwischen einem Musico theoretico und
einem Studioso musices (Lobenstein, 1748), p. 20.

3?ln Memoires de l'academie royale des sciences, 1758, pp. 856 f.
42

MEANTONE TEMPERAMENT

ment the "temperamento per coraune opinione perfettisimo,"38
and showed that it could be approximated by multiplying both
terms of the ratio 81:80 by 6 and then tempering the fifth by the
mean ratio 483:482, which gives 241:161 for the tempered fifth.
(A better approximation is 220:147.) From additional references
to the 55-division in Chapter VI, it would appear that this method
of tuning was in use for well over a century. As a system upon
which modulations might be made to any key, it was much better
than the 1/4-comma meantone system, although inferior to most
of the irregular systems discussed in Chapter VII.

Romieu mentioned temperaments of 1/7, 1/8, 1/9, and 1/10
commas, but did not consider them sufficiently important to dis-
cuss. The 1/10-comma temperament was included among Mar-
purg's many temperaments. 39 Otherwise none of these tempera-
ments has been advocated by any of our theorists. They should
be presented, however, in order to complete our study of regular
temperaments approaching equal temperament (see Tables
35-38). The syntonic comma has been divided in each case.

Table 35. 1/7 - Comma Temperament

2 +3 4+l 6 18 3 ,2 5

Names C° C#_1 D" Eb ' E~~ F " F#" G" G#" a" Bb 7 B~~ C°
Cents 0 92 198 303 396 501 593 699 791 897 1002 1095 1200
M.D. 6.3; S.D. 6.4

Table 36. 1/8 - Comma Temperament

7 l u+ 3 l + i 3 1 l 3 u+ 1 5

Names C° C#~a D" Eb ■ E" Fa F#" G" G#" A-"5 Bb * B_1 C°
Cents 0 95 199 302 397 501 596 699 794 898 1001 1097 1200
M.D. 4.0; S.D. 4.1

38Alessandro Barca, "Introduzione a una nuova teoria di musica, memoria
prima," Accademia di scienze, lettere ed arti in Padova. Saggi scientifici
e lettari (Padova, 1786), pp. 365-418.

3^f.W. Marpurg, Versuch iiber die musikalische Temperatur (Breslau,1776),
p. 163.

TUNING AND TEMPERAMENT

With the exception of some of Marpurg's symmetrical versions
of Neidhardt's unequal temperaments, the temperaments shown
in Tables 37 and 38 come closer to equal temperament than any
divisions that were not practical approximations to it.

Table 37. 1/9 - Comma Temperament

7 2 14 1 21 8125

Names C° C#" D" Eb+3 E" F"*"5 F#" g" G#" A" Bb » B"5 C°
Cents 0 97 199 301 398 500 598 700 797 899 1001 1098 1200
M.D. 2.3; S.D. 2.4

Table 38. 1/10 - Comma Temperament

Names C° C#" D" Eb+" E" F+^ F#"s G"^ G#" A_T5 Bb+s B" C°
Cents 0 99 200 301 399 500 599 700 798 899 1000 1099 1200
M.D. 1.2; S.D. 1.2

Chapter IV. EQUAL TEMPERAMENT

The first tuning rules that might be interpreted as equal tem-
perament were given by Giovanni Maria Lanfranco. * As stated,
these rules were for clavichords and organs (Monochordi &
Organi), but Lanfranco extended them also to the common stringed
instruments of his time. Thus there is none of the confusion that
arose later when the keyboard instruments were tuned in one
manner, the fretted instruments in another.

Lanf ranco's essential rules concern the tempering of the fifths
and the thirds: the fifths are to be tuned so flat "that the ear is
not well pleased with them," and the thirds as sharp as can be
endured. There seems to be a distinction here: for a fifth might
be tuned only slightly flat and the ear would not then be wholly
pleased with it; but the thirds are to be only a shade less harsh
than those which cannot be endured at all.

Most of Lanfranco 's contemporaries still knew no tuning but
the Pythagorean, with its pure fifths and impossibly sharp thirds.
Lanfranco's rules seem to represent a temperament of the Pytha-
gorean tuning, rather than of just intonation. Equal temperament
then fits his directions excellently. As further evidence, Lan-
franco divided the notes to be tuned into two classes, sharps and
flats. As with the meantone temperament, the sharps included
F#, C#, and G#, "although most of these are also common to the
flat class, if not in tuning, at least in playing." But, although the
flats proper included only BD and ED, this class "occasionally
needs in playing the black keys F* (Gb) and C# (D*3)." As Kin-
keldey says, "the enlargement of the major third, the diminution
of the minor third, the equivalence of the notes C# and DD, F*
and GD— these are essential departures from his contempor-
aries. "2

Aurelio Marinati^ honored Lanfranco by inserting in his "ex-

IScintille de musica (Brescia, 1533), p. 132.

2(Xto Kinkeldey, Orgel und Klavier in der I
77 f.

^Somma di tutte le scienza (Rome, 1587), pp. 95-98.

2(Xto Kinkeldey, Orgel und Klavier in der Musik des 16. Jahrhunderts, pp.
77f.

TUNING AND TEMPERAMENT

ample of the tuning of clavichords and organs" a word-for-word
account of Lanfranco's system, complete even to the title— with-
out, however, giving him credit for it. Another plagiarist, Cerone,
sufficiently appreciated Lanfranco to copy out his system for the
benefit of organ-builders. 4 At the time when these men were
writing, the meantone temperament was the recognized tuning
norm for keyboard instruments. It is rather surprising that
Cerone in particular, who had presented Zarlino's 2/7-comma
system in detail, did not seem to realize that there was a con-
flict between Zarlino's flat and Lanfranco's sharp major thirds.

Lodovico Zacconi^ was more astute. He presented no tuning
rules of his own, saying that it is "better that those who wish to
know and to see should look to the source and to the original
authors." For keyboard instruments he recommended Aron's
meantone temperament. "As for the other instruments, such as
the viole da braccio, viole da gamba, violins, and others, you can
look at the end of Giovanni Maria Lanfranco's book, which indi-
cates clearly how each one is to be tuned."

In Zacconi's day and long before it, the fretted instruments
were said to have equal semitones. To Zarlino, Salinas, and
Galilei this meant equal temperament, with all semitones equal.
To Grammateus and Bermudo, only ten semitones were equal,
the others being smaller; to Artusi, and presumably also to Bot-
trigari and Cerone, there were ten equal semitones, the other
two being larger. But, of these three types of temperament-
equal, modified Pythagorean, and modified meantone— only equal
temperament had both flat fifths and sharp thirds in addition to
equal semitones. Therefore, Zacconi, writing only sixty years
after Lanfranco, is practically saying that the latter 's rules rep-
resent equal temperament. In view of the excellent tuning
methods of Lanfranco's immediate predecessors, Grammateus
and Schlick, it is very likely that Lanfranco did intend equal tem-
perament for all instruments, including clavichords and organs.

Later writers who gave practical tuning rules for equal tem-
perament were often no more precise than Lanfranco had been.

4See Kindeldey, op. cit., p. 80.

5Prattica di musica (Venice, 1592), Part I, p. 218.

EQUAL TEMPERAMENT

Jean Denis, 6 for example, said nothing about the size of the
thirds. But all the fifths are to be lowered a trifle (d'un poinct),
"and all the fifths ought to be tempered equally." Denis may
even have had some variety of meantone temperament in mind,
for he directed that the tuning should begin with E*3 and end with
G#. But if his "toutes" means what it says, his was equal tem-
perament.

Godfrey Keller's tuning rules for harpsichord or spinet were
widely circulated, having been reprinted in the appendix to William
Holder's Treatise . . . of Harmony (London, 1731), and in Part VI of
Pierre Prelleur's long popular Modern Musick-Master.? Al-
though they can refer to nothing but equal temperament, they are
by no means accurate: "Observe all the Sharp Thirds must be as
sharp as the Ear will permit; and all Fifths as flat as the Ear
will permit. Now and then by way of Tryal touch Unison, Third,
Fifth, and Eighth; and afterward Unison, Fourth, and Sixth." It
is impossible for the thirds to be very sharp and the fifths simul-
taneously very flat; for in the 1/5-comma variety of meantone
temperament, in which the error of the fifths and the thirds is
equal, the error is not large. Keller's rules would read better
if he had said that the fifths were to be only slightly flat.

Barthold Fritz** gave tuning rules for equal temperament that
merited the approval of Emanuel Bach, to whom he had dedicated
his little book. Bach said that "in my [Fritz's] few pages every-
thing had been said that was necessary and possible, and that
would satisfy far more needs than the sundry computations with
which many a man has racked his brains; since the latter method

^Traite* de l'accord de l'espinette (Paris, 1650), pp. lOf.

"*Keller'sbook had the title A Compleat Method . . . (London: Richard Meares).
The British Museum has a copy dated 1707, but with a different printer.
The Library of Congress copy does not contain the tuning rules; its copy of
the Prelleur book is the 4th edition, dated 1738. The British Museum has an
edition of the latter dated 1731. Part VI was printed separately with the
title The Compleat Tutor for the Harpsichord or Spinet, and passed through
several editions, with various printers, in the 1750's and '60's.

°Anweisung wie man Claviere, Clavicins, und Orgeln, nach einer mechanis-
chen Art, in alien zwblf Tonen gleich rein stimmen konne, . . . (3rd edition;
Leipzig, 1780).

TUNING AND TEMPERAMENT

of instruction was only for very few people, but mine was for
everybody, the computers not excepted, because they depend upon
the judgment of the ear as well as the others. "9

Fritz's rules were very simple. After going from F to A by
four tempered fifths, he said, "I now have the already pure F as
a major third to this A, and, by touching the A and by testing it
with F, can hear whether it sounds sharp enough or so much up-
wards that the beats are about the rapidity of eighth notes in
common time. "10

Fritz began his tuning in the octave below middle C. From
William Braid White's table, ** the tempered F- A in this octave
will beat about 7 times per second, or over 400 times in a min-
ute. Even allowing for the somewhat lower pitch of the eighteenth
century, Fritz's eighth notes would be very fast, unless by "com-
mon time" he meant alia breve.

Mersennel2 also gave a practical tuning hint for equal tem-
perament when he said, "Certain people believe that they can find
the preceding accord of the equal semitones by beginning ut, re,
mi, fa, etc. on each key of the spinet, or by the number of trem-
blings or beats which the fifth and other tempered consonances
make: for example, the fifth beats once in each second when it
is tempered as it should be (as much for the organ as for the
spinet); whereas when it is just it does not beat at all." From
White's table, Mersenne's rule would apply best to the fifth D-A
in the octave above middle C, and approximately to other fifths
in that vicinity.

Alexander Ellis' practical rules for the formation of equal
temperament* 3 may be paraphrased as follows: If one tunes by
upward fifths and downward fourths within the octave above mid-
dle C, each fifth should beat once per second, and each fourth

^Ibid., Preface to 2nd edition.

IQlbid., p. 14.

11 Piano Tuning and Allied Arts (4th edition; Boston, 1943), p. 68.

l^Harmonie universelle (Paris, 1636-37), Nouvelles observations physiques
& mathematiques, p. 20.

13h. L. F. Helmholtz, Sensations of Tone (2nd English edition, translated by
Alexander J. Ellis; London, 1885), pp. 489 f.

EQUAL TEMPERAMENT

three times in two seconds. Ellis stated that if this rule is fol-
lowed accurately, the error for no pitch will be greater than two
cents. Again using White's useful table, we find that the mean
value of the beats of the tempered fifths in the C-C octave is
1.02 and of the tempered fourths, 1.47, proving that Ellis' rule
is correct.

White himself "lays the bearings" in the F-F octave, 14 just
as Fritz did. Since the ratio of a tempered fifth is approximately
3:2, one might suppose that he would advocate beating rates that
are 2/3 of Ellis' values: fourths once per second, and fifths
twice in three seconds. However, he recommends that the fifths
beat three times in five seconds, or 36 times per minute, and
suggests setting a metronome at 72, with the bell ringing at every
second tick. Since, from his own table, the mean value of the
beats of his tempered fifths is .68 rather than .60, he would get
better results from setting the metronome at 80.

Bossier's methodic for achieving equal temperament is rem-
iniscent of Aron's method for the meantone tuning. Aron, it may
be remembered, first tuned his major third pure and then tuned
equally flat the four fifths that were used in constructing the major
third. Bossier first divided the octave by ear into three equal
parts— C-E-G#-C. Then he tuned a group of four fifths, as C-G-
D-A-E, slightly flat, so that the last would give the sharp major
third already found. The method would be continued until the en-
tire octave was tuned. Having these first three notes fixed gave
him points of reference, so that he could never go far wrong.
But he realized that the human ear is fallible, for he recommended
that the tuner buy "steel forks from Frankfurt or Leipzig for all
twelve notes."

Geometrical and Mechanical Approximations

One of the famous problems of antiquity was the duplication
of the cube. It had been proved that the construction of the cube
root of 2 could not be accomplished by Euclidean geometry, that

14Op. cit., p. 85.

15H. P. Bossier, Elementarbuch der Tonkunst (Speier, 1782), pp. xxiv-xxvi.

TUNING AND TEMPERAMENT

is, by compass and ruler. This is the precise problem involved
in the solution of equal temperament by geometry, if Bossier,
for example, had desired to construct a monochord upon which
would be located his C-E-G#-C.

The first sixteenth century writer to suggest a geometrical
or mechanical means of solving equal temperament was Fran-
cisco Salinas. 16 Let him explain his method: "We judge this one
thing must be observed by makers of viols, namely, that the oc-
tave must be divided into 12 parts equally proportional, which 12
will be the equal semitones. And since they cannot accomplish
this by the 9th of the 6th book [the mean proportional construc-
tion] or by any other proposition of Euclid, it will be the task to
use the instrument which we said was called the mesolabium,
invented (as they believe) by Archimedes: by which they will be
able to obtain aline divided into as many equal parts as they wish.
We have not bothered to append the rule of its construction here,
because mention is made of its principle by Vitruvius in his 9th
book on architecture; from whom and from his expositors they
will be able to obtain the method of constructing it: for it is to
practical men for framing most matters not only useful, but well-
nigh indispensible."

The mesolabium had been previously advocated by Zar lino for
constructing his 2/7- comma meantone temperament, and later
Zarlino was to follow Salinas' lead in recommending it for equal
temperament. Hutton defined the word as follows: "Mesolabe,
or Mesolabium, a mathematical instrument invented by the an-
cients, for finding two mean proportionals mechanically, which
they could not perform geometrically. It consists of three paral-
lelograms, moving in a groove to certain intersections. Its figure
is described by Eutocius, in his Commentary on Archimedes.
See also Pappius, Lib. 3. "17

With the aid of a clear diagram (Figure B) James Gow^S has
explained the operation of the mesolabium as follows: "If AB,
GH be the two lines between which it is required to find two mean

16De musica libri VH, p. 173.

^Charles Hutton, Mathematical Dictionary (new ed.; London, 1815).

l^A Short History of Greek Mathematics (Cambridge, 1884; reprinted, New
York, 1923), pp. 245 f.

EQUAL TEMPERAMENT

proportionals, then slide the second frame under the firstandthe
third under the second so that AG shall pass through the points
C, E,at which the diameters of the second and third frames, re-
spectively, cease to be visible. Then CD, EF are the required
two mean proportionals."

F H

Fig. B. The Mesolabium (From James Gow, A Short History of Greek
Mathematics [c. 1884])

Although Zarlino contended that the mesolabium might be used
for finding any number of means, by increasing the number of
parallelograms, his diagram is for two means only. Of course
for equal temperament or for the 1/3-comma meantone temper-
ament, two means would suffice. But Salinas also advocates it
for an unlimited number of means, and Rossi would find the thirty
means for Vicentino's division by its aid. Mersenne,19 however,
in commenting upon Salinas' construction for equal temperament,
said it was incorrect if he intended to use the mesolabium for
more than two means, because the instrument mentioned by Vi-
truvius "is of no use except for finding two means between two
given lines." We shall not attempt to pass judgment upon these
conflicting opinions, but it would seem that the difficulty of the
process would be increased greatly with an increasing number
of means.

Zarlino^O has given three methods by which "to divide the
octave directly into 12 equal and proportional parts or semi-
tones." The first used the mesolabium, as already mentioned.
The second used the method of Philo of Bysantium (second cen-
tury, B.C.), which consisted of a circle and a variable secant

^Harmonie universelle, p. 224.

20Gioseffo Zarlino, Sopplimenti musicali (Venice, 1588), Chap. 30.

TUNING AND TEMPERAMENT

through a point on its circumference. The third is a variation of
the first, in that the string-length for one note is found by the
mesolabium, and then the lengths for the other notes are found
by similar proportions.

Mersenne,21 too, has contributed non-Euclidean methods for
finding two geometric means. The first, ascribed to Molthee,
used straight lines only, in the form of intersecting triangles.
The other method (Figure C) was furnished byRoberval and used

Fig

m 3

f A

|C ]

— ^

v u

\ y

\^ I

. C . Rob

erval's Method for Finding
Two Geometric Mean Pro-
portionals (From Mersenne's
HIarmonie universelle)

Reproduce
the Librai

>d by courtesy of
y of Congress

a parabola and a circle. 22 Kircher23 combined the Euclidean
method for finding one mean proportional with a mechanical
method for finding two means. This latter is by still another
method, consisting of two lines at right angles and two sliding

21Op. cit., p. 68.

22ibid, p. 408.

23Athanasius Kircher, Musurgia universalis (Rome, 1650), I, 207.

EQUAL TEMPERAMENT

L -shaped pieces, like carpenters' squares (Figure D). Accord-
ing to Rossi, 24 Kircher's is the method of Nicomedes, and Rossi
considered it "more expeditious" than others that have been men-
tioned. Marpurg25 ascribed Kircher's method to Plato, and
added methods by Hero and by Newton, together with Descartes'
method for finding any number of mean proportionals. Thus we
have more than half a dozen geometrical and mechanical methods,
proposed particularly for constructing a monochord in equal
temperament.

Fig. D. Nicomedes' Method for Finding
Two Geometric Mean Pro-
portionals (From Kircher's
Musurgia universalis)

Reproduced by courtesy of
the Library of Congress

Since these mechanical methods for finding two mean pro-
portionals are rather awkward, the attempt has been made to use
a satisfactory ratio for the major third or minor sixth, so that
the remainder of the division could be made by the Euclidean
construction for finding a single mean. Mersenne26 has given
two such methods. In the second, which he said is "the easiest
of all possible ways," the just value of the minor sixth (8:5) is
used. By mean proportionals, eight equal semitones are found

24sistema musico, pp. 95 f.

25yersuch liber die musikalische Temperatur, 19. Abschnitt.

26Harmonieuniverselle, p. 69.

TUNING AND TEMPERAMENT

between the fundamental and the minor sixth, and then, in like
manner, the remaining four semitones between the minor sixth
and the octave.

As can be seen from Table 39, this method is not extremely
close to correct equal temperament, because the just value of
the minor sixth is about 14 cents higher than its value in the equal
division. One might have expected the usually astute Mersenne
to have chosen a tempered value in the first place. The equally
tempered minor sixth is very nearly 100:63, as can be readily
seen in Boulliau's table given by Mersenne, where it bears ex-
actly this value. If this fraction is too difficult to work with,
27:17 will serve almost as well, and 19:12 comes rather close
also. Any of these other ratios would have given a more satis-
factory monochord than his. In Table 40, 19:12 is used for the
minor sixth.

Table 39. Mersenne's Second Geometrical Approximation

Names CxDxEFxGxA x B C

Cents 0 102 203 305 407 508 610 712 814 910 1007 1103 1200

M.D. 2.3; S. D. 2.5

Table 40. Geometrical Approximation (19:12 for Minor Sixth)

Names

C x

E F

Cents

0 99.5

198.9

298.4

397.8 497.3

596.7

696.2

Names

Cents

795.6

896.7

997.8

1098.9

1200.0

M.D. .76; S.D. .78

But we cannot be supercilious regarding Mersenne's other
practical method for obtaining two mean proportionals. Mer-
senne himself correctly said, "It serves for finding the mechan-
ical duplication of the cube, to about 1/329 part. "27 By the fa-
miliar Euclidean method he found the mean proportional between
a line and its double, subtracted the original line from the mean,

27lbid., p. 68.
54

EQUAL TEMPERAMENT

and then subtracted this difference from the doubled line. The
length thus found was the larger of the desired means— that is,
the string-length for the major third. In numbers, this ratio is
(3 - 42): 2, or .79289, which represents 401.8 cents. The result
is shown in Table 41, the remaining values being found by mean
proportionals as in Mersenne's second approximation. This is
an extremely fine geometrical way to approximate equal tem-
perament.

Table 41. Mersenne's First Geometrical Approximation

Names

C x

x E F

Cents

0 100.4

200.9

301.3 401.8 501.6

601.3

701.1

Names

x B

Cents

800.9

900.6

1000.4 1100.2

1200.0

M.D. .30; S.D. .32

Table 42. Ho Tchhe'ng-thyen's Approximation

Lengths 900 849 802 758 715 677 638 601 570 536 509.5 479 450

Names C C* D D* E E# F# G G# A A* B C

Cents 0 101 200 297 398 493 596 699 791 897 985 1091 1200

M.D. 4.8; S.D. 5.8

Numerical Approximations

The earliest numerical approximation for equal temperament
comes from China. About 400 A.D.,H6 Tchh^ng-thyen gave three
monochords for the chromatic octave, with identical ratios, but
with the fundamental taken as 9.00, 81.00, and 100.0 respec-
tively.28 (string-lengths are given for the first of these tables
only, since they illustrate the manner of its formation better than
the other two.)

Table 42 shows a remarkable temperament for the time when
it was constructed, comparable to the brilliant solution of the

^"Maurice Courant, "Chine et Core'e," Encyclopedic de la musique et diction-
naire du conservatoire (Paris, 1913), Part 1, Vol. I, p. 90.

TUNING AND TEMPERAMENT

problem of equal temperament by Prince Tsai-yu over a thousand
years later. At the time of Tchh^ng-thyenthe Pythagorean tuning
was the accepted system in China. If we assume the calculation
to begin with the higher C at 450 and proceed in strict Pythagorean
manner to B# in the lower octave, the B^ will be at 888 instead
of 900. This is 12 units too short. Let us, therefore, add 1 unit
to 600, the value for G; 2 units to 800, the value for D; 3 units
to 533, the value for A; and so forth, along a sequence of fifths,
until we reach the correct value for C at 900. Tchh§ng-thyen's
figures agree precisely with our hypothesis.

A linear correction, such as Tchh§ng-thyen made, often pro-
vides a good approximation, as we shall see elsewhere in this
chapter. The difficulty with his correction is that if he had started
with the lower C and had continued until he had reached the higher
B^, the latter would have been only 6 units too short instead of 12.
By adding 10 parts for A#, 8 for G^, etc., he obtained pitches that
were much too low. If he had added 12 parts to 444 for the higher
B^, the corrected length, 456, would have been at 1177, instead
of 1200 cents, 23 cents flat! Let us consider the effect of adding
precisely half the correction for each note. This would work
well for the odd semitones, C D E F# G# A# B#, as might have
been expected; but the lower three even semitones, C^ D# E^,
are then as sharp as the higher odd semitones were flat before!
We shall have better success if we continue the series of whole
tones from G to Fx, the latter at 296 needing a correction of 4.2
to make a perfect octave to G, 600.5. Then the intermediate
notes can be given a proportional linear correction, which would
be doubled for the three notes C* D# E# when transposed to the
lower octave. This improved temperament is shown in Table 43.
The greatest error is at C*.

Table 43. H6 Tchh£ng-thyen's Temperament, Improved
Lengths 900 846.6 801 754.8 713 763 635 600.5

Names

D x E

Cents

106

202 305 403

503

604

701

Lengths

566

534.1

504.5 475.7

450

Names

x B

Cents
6

803

903

]

1004 1101
M.D. 2.2; S.D. 2.7

1200

EQUAL TEMPERAMENT

The arithmetical division of the 9:8 tone into 17:16 and 18:17
semitones was known to all sixteenth century writers through
Ptolemy's demonstration that Aristoxenus could not have obtained
equal semitones in this way. But Cardano (1501-76) may have
been referring to some practical use of the 18:17 semitone when
he wrote: "And there is another division of the tone into semi-
tones, which is varied by putting the tone between 18 and 16; the
middle voice is 17; the major semitone is between 17 and 16, but
the minor between 18 and 17, the difference of which is 1/288.
It is surprising how the minor semitone should be introduced so
pleasingly in concerted music, but the major semitone never. "29

The simplest way to construct a monochord in equal tempera-
ment is to choose a correct ratio for the semitone and then apply
it twelve times, a construction that can be performed very easily
by similar proportion. Vincenzo Galilei^O must be given the
credit for explaining a practical, but highly effective, method of
this type. For placing the frets on the lute he used the ratio 18:17
for the semitone, saying that the twelfth fret would be at the mid-
point of the string. He went on to say that no other fraction would
serve; for 17:16, etc., would give too few frets, and 19:18, etc.,
too many. Since 18:17 represents 99 cents, 17:16, 105 cents, and
19:18,94 cents, Galilei was correct in his contention. But he did
not give a mathematical demonstration of his method. It remained

Table 44. Galilei's Approximation

Lengths

100000

94444

89197

84242

79562

75142

70967

Names

Cents

198

297

396

495

594

Lengths

67024

63301

59784

56463

53326

50000

Names

Cents

693

792

891

990

1089

1200

M.D. 1.8: S.D. 3.3

29Girolamo Cardano, Opera omnia, ed. Sponius (Lyons, 1663), p. 549.
30Dialogo della musica antica e moderna (Florence, 1581), p. 49.

TUNING AND TEMPERAMENT

for him a proof by intuition. The string- lengths in Table 44 were
calculated by Kepler. 31

Mersenne32 testified that Galilei's method was favored by
"many makers of instruments." The Portugese writer Domingos
de S. Jose Varella33 gave a "way to divide the fingerboards of
viols and guitars." This is precisely Galilei's method, and
Varella told how the construction could be continued by similar
proportion after the first 18:17 semitone had been formed. Like-
wise Delezenne34 showed that 18:17 is very near the value for
the correct equal semitone, and gave a geometrical construction
for it used by Delannoy, the instrument maker, in placing the
frets upon his guitars.

Two other early nineteenth century references to what Gar-
nault35 called the "secret compass" of the makers of fretted
instruments were given in his tiny and not very trustworthy
monograph on temperament. The first was from the Robet-
Maugin Manuel du Luthier (1834), which stated that if the string
is 2 feet in length, the first semitone will be at a distance of 16
lines from the end; this represents 16/2x12x12 = 1/18 the length
of the string, thus giving 18:17 for the ratio of each semitone.

Garnault's second reference was to the Bernard Romberg
'cello method (1839 ),36 which he said had been adopted byCheru-
bini for use in the Paris Conservatoire. Romberg's directions

"^Johannes Kepler, Harmonices mundi (Augsburg, 1619; edited by Ch. Frisch,
Frankfort am Main, 1864), p. 164.

^^Harmonie universelle, p. 48.

33compendio de musica (Porto, 1806), p. 51.

34c E. J. Delezenne, "Memoire sur les valeurs numeriques des notes de la
gamme," Recueil des travaux de la soci^te* des sciences, ... de Lille,
1826-27 p. 49, note (a), and p. 50.

35paul Garnault, Le temperament, son histoire, son application aux claviers,
aux violes de gambe et guitares, son influence sur la musique du xviiie
siecle (Nice, 1929), pp. 29 ff.

36in the German translation (original?), Violoncell Schull (Berlin, 1840 [?] ),
the directions are given on page 17; in the English translation, A Complete
Theoretical and Practical School for the Violoncello, they are omitted.

EQUAL TEMPERAMENT

were much the same as those given previously. Although Gar-
nault does not mention this, Romberg added that the directions
given were for equal temperament, but the more advanced player
would often make the sharped notes sharper and the flatted notes
flatter than these pitches— another confirmation of the quasi-Py-
thagorean tuning of instruments of the violin family.

These references to the 18:17 semitone cover two and a half
centuries. It is probable that they could be brought much nearer
our own times if the makers of fretted instruments were, given a
chance to express themselves. We must accept Galilei's method,
therefore, as representing the contemporary practice. A player
on a lute was not going to bother with the mesolabium or with a
monochord on which were numbers representing the successive
powers of the 12th root of 2. But he could place his frets by a
simple numerical ratio such as 18:17, and we are glad that the
frets thus placed served their purpose so well.

Critics of Galilei were not slow to show that the 12th fret
would not coincide precisely with the midpoint of the string.
Passing by the inconveniently large numbers of Zarlino's ratios,
we come to Kepler's result: if the entire string is 100,000 units
in length, Galilei's 12th fret will be at 50,363 instead of 50,000.
As we have already stated, his semitone has only 99 cents, so
that the octave contains 1188 instead of 1200.

There are various ways of correcting the octave distortion
arising from the use of the 18:17 semitone. An obvious way is
suggested by Mersenne's approximations: form only 4 semitones
with the 18:17 ratio; then apply Mersenne's mean-proportional
method to the remaining 8 semitones. The monochord thus con-
structed (Table 45) is as good as Mersenne's first method.

Table 45. Approximation a la Galilei and Mersenne

Names

Cents

198

297

396

496.5

597

697.5

Names

Cents

798

898.5

999

1099.5

1200

M.D. .67; S.D. .71

TUNING AND TEMPERAMENT

An even simpler correction uses linear divisions only: since
the length for the 12th fret is 363 units too great, divide 363 into
12 equal parts and subtract 30 units for the first fret, 61 for the
second, 91 for the third, etc. As is always the case with this
type of correction, there is a slight bulge in the middle of the
octave, but the largest error is only 1.8 cents.

The correction shown in Table 46 lends itself well to numer-
ical computation, since the fundamental and its octave are in
round numbers. But in practice, with a geometrical, not a numer-
ical, construction, the following would be simpler and is even a
trifle better: if 50,363 be considered the real middle of the
string, the octave will be perfect. To make it the middle, shorten
the entire string by twice the difference between 50,000 and
50,363, that is, by 726. Then everyone of the lengths as given by
Kepler will be diminished by 726, and the 12th fret, 49,637, will
be the exact middle of the string, 99,274. Note again the slight
bulge in the middle of the division (Table 47), with the greatest
distortion 1.0 cent.

Table 46. Galilei's Temperament, with Linear Correction, No. 1

Lengths

100000

94414

89136

84151

79441

74991

70785

Names

Cents

99.5

199.1

298.8

398.5

498.3

598.2

Lengths

66812

63059

59512

56160

52993

50000

Names

Cents

698.3

798.4

898.5

998.9

1099.4

1200

M.D. .26; S.D. .31

Table 47. Galilei's Temperament, with Linear Correction, No. 2

Lengths

100000

99274

88471

83516

78836

74416

70241

Names

Cents

99.7

199.4

299.3

399.1

499.0

599.0

Lengths

66298

62575

59058

55737

52600

49637

Names

Cents

699.0

799.0

899.2

999.3

1099.7

1200

M.D. .17; S.D. .21

EQUAL TEMPERAMENT

The improvements upon Galilei's tuning shown in Tables 46
and 47 could have been made by practical tuners. They are better
divisions than many of the numerical expressions of equal tem-
perament which will be shown later. They are better also than
the temperament our contemporary tuners give our own pianos
and organs. So there is nothing more that needs to be said, as
far as practice is concerned. There are, however, several other
and more subtle ways of improving Galilei's tuning which we
should like to mention. These are of speculative interest solely.

Letus return to the false octave generated by the 18:17 semi-
tone. Mersenne suggested that "if the makers should increase
slightly each 18:17 interval, they would arrive at the justness of
the octave." The 11th fret is at 53326, leaving a ratio of 53326:
50000 for the remaining semitone. This, as its cents value in-
dicates (111 cents), is about the size of the just 16:15 semitone.
Let us pretend that the final digit in the antecedent is 5, and re-
duce the ratio to 2133:2000. Now let us average this semitone
with the eleven 18:17 semitones, using the arithmetical division
generally followed by sixteenth century writers. Our desired
semitone is 2000/2133 + 187/18 = 48319. In decimal form this

12 51192

is .9438779, as compared with the true equal semitone, .9438743.
The successive powers of this decimal would deviate more and
more from those of the 12th root of 2, but even then the octave
would be only .1 cent flat.

Another way of correcting Galilei's tuning is based upon the
fact that his octave would be 12 cents, that is, half a Pythagorean
comma, flat. A somewhat crude, but practical, manner of ad-
justing the octave would be to form four 18:17 semitones, from
C to E, then take the next five notes, F through A, as perfect
fourths to the first five, and then the two remaining notes, B*3
and B, as perfect fourths to F and F#. A satisfactory monochord
is shown in Table 48. Note particularly how much smaller its
standard deviation is than that of Galilei's actual tuning.

As an approach to a finer division using Pythagorean inter-
vals, let us turn to Pablo Nassarre.37

37Escuela musica (Zaragoza, 1724), Part I, pp. 462 f.

TUNING AND TEMPERAMENT

Table 48. Galilei's Temperament Combined with Pythagorean

Names

Cents

198

297

396

498

597

696

Names

Cents

795

894

996

1095

1200

M.D. 1.5; S.D. 1.6

He had discussed equal
semitones upon fretted instruments, using much the same lan-
guage as Praetorius,38 to the effect that a 16:15 diatonic semi-
tone contains 5 commas and a 25:24 chromatic semitone 4 com-
mas, but that these semitones have the peculiarity that they are
all equal, containing 4 1/2 commas. They are obtained by a
linear division of the 9:8 tone into 18:17 and 17:16 semitones.
To place the frets, three or four 9:8 tones are constructed, and
the distance between each pair of frets divided equally to form
the semitones . Of course an arithmetical division of tones will
not form precisely equal semitones. Furthermore, there is a
fairly large distortion for the last semitone if the process is
carried out through twelve semitones. Of course, as with
Galilei's method, no single string would have had twelve frets.
In Table 49 the division is made for the entire octave. The length
for B was taken as the arithmetical mean between A^ and the
middle of the string.

Table 49. Nassarre's Equal Semitones

Names

C°

D°

E°

(F)

F#>

(G)

Cents

204

303

408

507

612

711

Names

G#°

(A)

A*0

(B)

C°

Cents

816

915

1020

1107

1200

M.D. 4.2; S.D. 5.4

If Nassarre had divided each 9:8 tone into precisely equal
semitones by a mean proportional, his errors would have been
smaller.

•^Syntagma musicum, Vol. 2, p. 66.
62

EQUAL TEMPERAMENT

Table 50. Nassarre's Temperament Idealized

Lengths

100000

94281

88889

83805

79012

74494

70233

Names

c°

D°

E°

(F)

F#°

Cents

102

204

306

408

510

612

Lengths

66216

62429

58859

55493

52319

50000

Names

(G)

G*°

(A)

A*>

(B)

C°

Cents

714

816

918

1020

1110

1200

M.D. 3.7; S.D. 6.7

It is not particularly difficult to set down this tem-
perament in figures, since the square root need be performed
only for C*, after which a second series of 9:8 tones can be
formed, starting with this note. If B is taken as the geometric
mean between A^and C, its length is 52675, or 1110 cents, making
the mean deviation 3.3, and the standard deviation 4.5. However,
for the sake of an approximation to be made in Table 50, B is
taken as the geometric mean between A^ and B^, with a relatively
high standard deviation.

If we now compare the cents values of the temperament shown
in Table 50 with those of Galilei's tuning, we shall find that the
error of the former is opposite to and twice as great as that of
the latter. Therefore, for every pair of string- lengths, subtract
the smaller (Nassarre) from the larger (Galilei), and then sub-
tract 1/3 the difference from the larger number. The excellent
monochord shown in Table 51 results.

Table 51. Temperament a la Galilei and Nassarre

Lengths

Names

Cents

Lengths

Names

Cents

100000

0
66755

G
699.7

94390

99.9

63010

x
799.7

89094

D
199.9
59476

A
899.6

84096

x
299.9
56140

x
999.6

79379

E
399.8
52990

B
1099.6

74926

F
499.8
50000

C
1200

70722

x
599.7

M.D. .07; S.D. .13

TUNING AND TEMPERAMENT

If the idealized Nassarre temperament had been extended one
more semitone, the string-length for the octave would have been
49,328. When this number is adjusted with the 50,363 of Galilei's
tuning, the octave proper to the above temperament becomes
50,018 or 1199.5 cents. Let us now make the same type of octave
adjustment as with the original Galilei tuning, by subtracting 18
from the 12th semitone, and 1 or 2 less for each succeeding
semitone. Then no length varies by more than 2 or 3 units from
the correct value, that is, the maximum variation is less than .1
cent.

This procedure sounds somewhat complicated. It is not nec-
essary to go through the entire process three times, as shown
above, in order to obtain the final monochord. The ratio for the
semitone will be 17/9 + 2^2~/3 = 17 + 6/J2~ . Including the octave

3 27

correction, the formula for the string-length of the nth semitone
is: 100,000 /l7 + 6^ \- 3(n-l) . Perhaps it woukfbe simpler

27/2

after all to stick to cube roots, especially when fortified with a
table of logarithms !

Johann Philipp Kirnberger,39 however, used a very rounda-
bout method of attaining equal temperament, believing it to be
simpler in practice than tuning by beating fifths. He showed that
the ratio 10935:8192 closely approaches the value of the fourth
used in equal temperament. In practice this value would be ob-
tained by tuning upward seven pure fifths and then a major third.
In other words, if C° is the lower note, E^_1 is regarded to be
the equivalent of F~^ , the tempered fourth. The basis for this
equivalence lies in the fact that the schisma, the difference be-
tween the syntonic and the ditonic commas, is almost exactly
1/12 ditonic comma, the amount by which the fourth must be
tempered. The ratio given above becomes, in decimal form,
.7491541 . . . , whereas the true tempered value is .7491535 ....
The result is an extremely close approximation.

39Die Kunst des reinen Satzes in der Musik, 2nd part (Berlin, 1779), 3rd Di-
vision, pp. 179 f.

EQUAL TEMPERAMENT

Kirnberger spoke of Euler's approval of his method, and of
Sulzer's and Lambert's publication of it. Marpurg^O showed
that Lambert's method, when applied to an entire octave, will
differ for no note by more than .00001. He praised it as a method
that needs no monochord, and believed that the tuning of the just
intervals used in it could be made more quickly and accurately
than the estimation by ear of the tempering needed for the fourth
or the fifth. However, the tuning of a pure major third is so dif-
ficult that Alexander Ellis thought that better thirds can be ob-
tained from four beating fifths than by tuning the thirds directly.
If this be true, a type of tuning in which the essential feature is
a pure major third could not be very accurate, without consider-
ing the labor of tuning eight pure intervals in order to have only
one tempered interval!

Kirnberger 's approximation for equal temperament was next
heard of in England, where John Farey^l seems to have dis-
covered it independently. In Dr. Rees's New Cyclopedia^ 2 we
are shown how Farey's method "differs only in an insensible
degree" from correct equal temperament.

Among the monochords shown by Marpurg is one by Daniel P .
Strahle,43 allegedly in equal temperament, but actually unequal,
as can be seen in Table 52. This is a geometric construction of
a curious sort, for which Jacob Faggot computed the string-
lengths by trigonometry (see Figure E). In brief, it went like
this: upon the line QR, 12 units in length, erect an isosceles tri-
angle, QOR, its equal legs being 24 units in length. Join O to the
eleven points of division in the base. On QO locate P, 7 units
from Q, and draw RP, extending it its own length to M. Then if
RM represents the fundamental pitch and PM its octave, the

^Oyersuch liber die musikalische Temperatur, p. 148.

41uOn a New Mode of Equally Tempering the Musical Scale," Philosophical
Magazine, XXVII (1807), pp. 65-66.

42ist American edition, Vol. 14, Part 1, article on Equal Temperament.

43«Nytt pafund, til at finna temperaturen, i stamningen for thonerne pa cla-
veretock dylika instrumenter," Proceedings of the Swedish Academy, IV
(1743), 281-291. .The second part of the article, "Trigonometriskutrakning,"
appears under Faggot's name.

TUNING AND TEMPERAMENT

points of intersection of RP with the 11 rays from O will be the
11 semitones within the octave.

Table 52. Faggot's Figures for Strahle's Temperament

Lengths

10000

9379

8811

8290

7809

7365

6953

Names

Cents

111

219

325

428

529

629

Lengths

6570

6213

5881

5568

5274

5000

Names

Cents

727

824

919

1014

1108

1200

M.D. 4.8; S.D. 5.7

Fig. E. Strahle's Geometrical Ap-
proximation for Equal
Temperament
Reproduced by
courtesy of the
Library of the
University of
Michigan

It is obvious from the construction that the distance between
two consecutive points of division will be greater near R than
near P, and hence that, superficially at least, the division will
resemble a series of proportional lines, as in true equal tem-
66

EQUAL TEMPERAMENT

perament. But, as Table 52 shows, there is a large bulge in the
middle of the octave, and F*, which should be 5000^2 = 7071, is
distorted very greatly. Now, if QR is given, the points of division
are functions of QO (or RO), but they are also functions of QP.
It is primarily the size of the angle QRP that determines the
ratios of the string- lengths. Strahle's choice of 7 units for QP
was unfortunate, or the distortion would not have been so great.
To reduce the errors in this construction, let us attempt to
find a value for the angle QRP for which the length for F# is
correct, V2RM. Let A be the midpoint of QR and B the point

2
where OA cuts RM; so that BM is the length for F- . Then

1. RB = 42BP = a(2RP

1W2~

2. OQR = cos-1 1/4 = 750 31' .
By the sine law and from 1. and 2.,

3. sin RPQ 12, or sin RPQ 12

sin PQR ~~ RP fl5/4 RB/1 + a/2"

4. cos QRP = 6/RB.
From 3. and 4.,

5. sin RPQ = ^30 cos QRP 1.1344 cos QRP

2(1+^2) "

From 2.,

6. QRP + RPQ = 104° 29' .
As an approximate solution to 5. and 6.,

7. QRP = 33° 36' and RPQ = 70° 53'.

From 7., PQ = 7.028. But this is almost exactly Strahle's
figure! A check reveals that Faggot made a serious error in
computing the angles QRP and RPQ; so that his value for PQ
was actually 8.605 rather than 7. Table 53 gives the correct
figures for Strahle's temperament.

TUNING AND TEMPERAMENT

Table 53. Correct Figures for Strahle's Temperament

Lengths

100000

9432

8899

8400

7931

7490

7073

Names

Cents

101

202

302

401

500

600

Lengths

6676

6308

5955

5621

5303

5000

Names

Cents

699

798

897

997

1098

1200

M.D. .83; S.D. 1.00

It is, therefore, possible to achieve superfine results by fol-
lowing a method essentially the same as Strahle's. Although un-
aware of the possibilities in Strahle's method, Marpurg has col-
lected many unusual and interesting temperaments by other
men. 44 Represented two monochords by Schrbter,both of which
are excellent approximations to equal temperament constructed
from tabular differences. In the first (Table 54), Schroter an-
chored his column of differences upon the notes of the just minor
triad, as C ED G C, with ratio 6:5:4:3. The intermediate notes
were obtained by arithmetical divisions. This column of differ-
ences is worth showing as a monochord in its own right, for the
method of construction resembles that of Ganassi and Reinhard.
The mean deviation is about the same as for the Pythagorean
tuning, but the standard deviation is larger because the semitone
B-C, with ratio 28:27, is much smaller than the others.

Table 54. Schroter 's Column of Differences, No. 1

Lengths

Names

Cents

204

317

435

520

608

702

Lengths

Names

Cents

804

906

1018

1137

1200

M.D. 11.9; S.D. 15.3
44yersuch uber die musikalische Temperatur, pp. 179 ff.

EQUAL TEMPERAMENT

In Schrbter's monochord proper (Table 55) the upper funda-
mental (451) is the sum of all the differences in the above table,
save the first number to the left (54). Thus the lower fundamental
(902) will be a true octave. This monochord is a highly satis-
factory approximation to equal temperament.

Table 55. Schroter's Approximation, No. 1

Lengths

902

851

803

758

716

676

638

Names

Cents

100.7

201.3

301.1

399.9

499.3

599.7

Lengths

602

568

536

506

478

451

Names

Cents

700.0

800.7

901.1

1000.8

1099.4

1200

M.D. .52; S.D .59

Schroter's column of differences for the second approximation
(Table 56), while also containing arithmetical divisions, is con-
structed more carefully than the first. The minor thirds D-F
and A-C have the unusual ratio 19:16 or 297 cents. All the notes
in the tetrachord G-C are pure fifths above the notes inthetetra-
chord C-F. Here the deviation is about the same as in Gram-
mateus' tuning, thus ranking among the best of the irregular sys-
tems .4 5

Table 56. Schrbter's Column of Differences, No. 2

Lengths

384

363

342

324

306

288

272

256

242

Names

Cents

201

294

393

498

597

702

799

Lengths

228

216

204

192

Names

Cents

903

996

1095

1200

M.D. 3.8; S.D. 4.3

4^For Grammateus see the second part of Chapter VII.

TUNING AND TEMPERAMENT

Schroter's second approximation (Table 57) is constructed
from the above column of differences in the same manner as was
his first. Its deviations, like those of the column of differences
upon which it was based, are about 1/3 as large as those of the
first monochord.

4843

600.3

Lengths

6850

6466

6103

5761

5437

5131

Names

Cents

99.9

199.9

299.7

400.0

500.2

Lengths

4571

4315

4073

3845

3629

3425

Names

Cents

700.3

800.1

900.0
M.D. .

15;

999.7
S.D.

1099.9

1200

Schroter's success in building up a monochord by using well-
chosen tabular differences suggests that the same method be
applied to Ganassi's tuning, which is rather similar to his first
column of differences .46 The sum of the twelve numbers of
Ganassi's monochord is 805, which is chosen, therefore, for the
higher fundamental. As might have been expected, the mono-
chord (Table 58) is very good.

Table 58. Approximation Based on Ganassi's Monochord

Lengths

1610

1520

1435

1355

1279

1207

1139

Names

Cents

99.6

199.3

298 6

398.5

498.8

599.2

Lengths

1075

1015

958

904

853

805

Names

Cents

699.3

798.8

898.8

999.3

1099.9

1200

M.D. .42; S.D. .51

46see Chapter VII for Ganassi's tuning.
70

EQUAL TEMPERAMENT

Table 59. Monochord from Difference Column, No. 1

Lengths

20 19

Names

E F

Cents

151

232

316 405

498

597

Lengths

Names

Cents

815

933

1062

1200
M.D. 18.2;

S.D. 19.7

702

These rather amusing improvements in poor or fair tuning
systems suggest that the method be really put to the test by
choosing for the original monochord an entirely unsatisfactory
tuning. Accordingly, the thirteen numbers from 12 through 24
were chosen (Table 59). This is so perverted a tuning system
that the major third (E), the fourth (F), and the fifth (G) are pre-
cisely a semitone flat according to just intonation. However, a
benighted anonymous writer in the Mercure de France in 1771
declared that if the entire string were divided into 24 parts, the
numbers 12 through 24 would give all the semitones .47 Thanks
to the regularity of its construction, the deviation of this system
ranks it somewhere near the meantone tuning!

In the next monochord (Table 60) the deviation is of the same
class as that of Galilei's tuning. Its higher fundamental, 210, is
the sum of the numbers 12 to 23 inclusive.

Table 60. Monochord from Difference Column, No. 2

Lengths 420 397 375 354 334 315 297

Names C x D x E F x

Cents 0 97.5 196.2 296.0 397.7 498.1 599.9

Lengths 280 264 249 235 222 210

Names G x A x B C

Cents 702.0 803.9 905.2 1005.4 1103 9 1200

M.D. 1.6; S.D. 1.9

47Lionel de La Laurencie, Le violon de Lullya Viotti (Paris, 1924), Tome HI
p. 74.

TUNING AND TEMPERAMENT

For our third monochord (Table 61) we use the lengths of
Table 60 as differences. Here the deviation is about the same
as in Schrbter's second approximation.

In the fourth and last approximation (Table 62) the errors
have become too small to be recorded correctly when five-place
logarithms are used. Apparently, however, the deviation is again
about 1/10 that of the previous monochord.

Table 61. Monochord from Difference Column, No. 3

Lengths

7064

6667

6292

5938

5614

5289

4992

Names

Cents

100.1

200.2

300.6

400.9

501.0

601.1

Lengths

4712

4448

4199

3964

3742

3532

Names

Cents

701.0

800.9

900.6
M.D. .18;

1000.3
S. D. .21

1100.1

1200

Table 62. Monochord from Difference Column, No. 4.

Lengths

118758

112091

105799

99861

94257

88968

83976

Names

Cents

100

200

300

400

500

600

Lengths

79264

74816

70617

66653

62911

59379

Names

Cents

700

800

900

1000

1100

1200

Objection may be made to Schroter's approximations, and to
ours as well, on the ground that the fundamentals are not round
numbers such as most of the theorists used for the representa-
tion of equal temperament. Let us see whether we can supply
this lack. In our third monochord (Table 61) the length for F#
is 4992. Let this be our higher fundamental. Add 8 to it, and 16
to its double, the lower fundamental. We could then make an
arithmetical division to correct the intermediate numbers. It is
little more trouble, however, to take the two left-hand digits of
the numbers in this same monochord, starting with the value for

EQUAL TEMPERAMENT

BD, 40. Multiply these and those for B, 37, by .4, as 16.0, 14.8,
and all the pairs of digits to the left of BD by .2. Add these num-
bers to the appropriate numbers in Monochord No. 3, and we have
a corrected monochord, in which the maximum error is 4 units,
or about 1 cent (see Table 63). Deviation is as in the original
Monochord No. 3 (Table 61).

Table 63. Monochord No. 3, Adjusted

Lengths

10000

9439

8910

8411

7940

7496

7075

Names

Lengths

6678

6302

5947

5613

5297

5000

Names

Fortunately, it is possible to make a similar adjustment of
our five-digit monochord, No. 4 (Table 62). Here we shall take
as our lower fundamental the length for ED, 99861. We need 139
to make a round number. This is about twice the length for G in
Monochord No. 2. So we divide the numbers in the second mono-
chord by 2 or by 4, and add to the appropriate numbers in Mono-
chord No. 4. The maximum error is 6 units, or about 1/6 cent.

A very useful approximation for equal temperament is to ex-
press all its irrational ratios as comparatively small fractions.
Alexander Ellis^S has made a table of about 150 intervals within
the octave, which he has represented by logarithms, cents, and
ratios, actual or approximate. Since all the intervals of equal
temperament are contained in this table, it is easy to list them
separately, as in Table 65.

Table 64. Monochord No. 4, Adjusted

Lengths 100,000 94,388 89,092 84,093 79,375 74,921 70,716

Names C x D x E F x

Lengths 66,747 62,999 59,462 56,124 52,974 50,000

Names G x A x B C

48H. L. F. Helmholtz, Sensations of Tone, pp. 453-456.

TUNING AND TEMPERAMENT

Table 65. Ellis' Fractional Approximations

Ratios 1 89:84 449:400 44:37 63:50 303:227 140:99 433:289

Names Cx D xEF x G

Ratios 100:63 37:22 98:55 168:89 2

Names x A x B C

Charles Williamson^ has given the material in Table 65,
wrongly ascribing it to Helmholtz rather than to Ellis. By con-
tinued fractions he himself found that the majority of Ellis' ratios
were correct. He objected to the ratio for the major second
(449:400), stating that this interval can be represented more ac-
curately as the inversion of a minor seventh. The ratios for the
fourth (303:227) and fifth (4 33: 289) he thought were not sufficiently
close either, and should likewise be paired. Ellis' ratio for the
tritone (140:99) was good, but Williamson preferred to use the
ratio for its inversion (99:70), which is no better.

Williamson remarked that his ratio for the tone (55:49) oc-
curs in Cahill's patent for the Telharmonium, and for the tritone
(99:70) in Laurens Hammond's patent for the Hammond Electric
Organ. He had not previously run across 295:221 or 442:295. It
is interesting to note that here, as in many other instances, Pere
Mersenne^O has anticipated the modern students of temperament.
Mersenne stated that the minor third of equal temperament is
approximately 6/5 x 112/113 = 672/565. Convergents to this
ratio are 44:37 and 157:132, the first of these occurring in both
tables above. Mersenne 's ratio for the major third was 5/4 x
127/126 = 635/504, convergents to which are 63:50 (as above)
and 286:227. For the perfect fifth he gave the ratio 32 x 886/887 =
1329/887, the convergent to which is 442:295, used by Williamson.

Williamson's reference to Hammond's patent^! suggests that
the latter 's ratios be examined in their entirety. (It must be
remembered that these ratios are based on the practical con-

49 "Frequency Ratios of the Tempered Scale," Journal of the Acoustical So-
ciety of America, X (1938), 135.

^^Harmonie universelle, Nouvelles observations physiques & mathematiques,
pTllL

51L. Hammond's Patent, 1,956,350, April 24, 1934, Sheet 18.
74

EQUAL TEMPERAMENT

sideration of cutting teeth on gears.) The difficulty is that, al-
though it is easy enough to reduce Hammond's frequencies to
ratios with no more than two digits in numerator and denomin-
ator, no one note appears as unity. (The ratios times 320 are
the frequencies from middle C to its octave.) We cannot well
compare this with Table 65. If either F or A, which have the
simplest ratios in Table 66, is given the value of 1, more than
half of the ratios will have three digits. Hence the composite
table, Table 67, with decimal equivalents, gives a better idea of
how the three systems compare.

Table

. Hammond's Fractional Approximations

Ratios 85:104

71:82

67:73

35:36

69:67

12:11 37:32

Names C

F x

Ratios 49:40

48:37

11:8

67:46

54:35

85.52

Names G

Table

. Compar

ison of Three Approximations

Ellis

Williamson

Hammond

Equal Temperament

C 200000

200000

B 188652

188652

188697

188775

x 178182

178182

178180

A 168182

168182

168179

x 158730

158730

158677

158740

G 149827

149831

149796

149831

x 141414

141429

141414

141421

F 133480

133484

133499

133484

E 126000

126000

125942

125992

x 118919

118919

118881

118921

D 112250

112245

112207

112246

x 105952

105952

105928

105946

C 100000

100000

Hammond has utilized some of the same ratios as Ellis and
Williamson. His tone G-A is 55:49; his minor thirds F-Ab and
F#-A are 44:37; his major third Eb-G is 63:50; his tritones
ED-A and F-B are 99:70. He had another major third (Bb-D)

TUNING AND TEMPERAMENT

with small ratio, 73:46, but this is a poorer approximation than
63:50. Note that many of Hammond's ratios are related in pairs,
but not in the same way as Williamson's. The product of the
ratios for F^ and G#, F and A, E and BD, and B and D^ is equal
to 3:2. C and D are not so related. Of course the axis G is ap-
proximately the square root of 3:2, and C*, the other axis, the
square root of 3:4.

Let us compare these three approximations with the true
values for equal temperament to six places (see Table 67). For
Ellis and Williamson these are the decimal equivalents of the
fractions as given. For Hammond the note A was taken as the
fundamental, and his frequencies as given in the patent have been
divided by 1.1.

In our absorption with quasi- equal temperaments that excel
many presumably correct versions, we should not neglect the
pioneers who first set down in figures the monochords constructed
upon the 12th root of 2. The first European known to have formed
such a monochord is Simon Stevin,52 about 1596, who said that
since there are twelve proportional semitones in the octave, the
problem is to "find 11 mean proportional parts between 2 and 1,
which can be learned through the 45th proposition of my French
arithmetic." There he had explained that mean proportionals can
be found by extracting roots of the product of the extremes. He
now applied this principle, by representing each semitone as the
12th root of some power of 2 (see Table 68).

Table 68. Stevin's Monochord, No. 1

Lengths

10000

9440

8911

8408

7937

7493

7071

Names

Cents

99.7

199.6

300.2

400.0

499.6

600.0

Lengths

6675

6301

5945

5612

5298

5000

Names

Cents

699.8

799.6

900.3

1000.1

1099.7

1200

52Van de Spiegeling der Singconst, pp. 26 ff.
76

EQUAL TEMPERAMENT

In his actual calculations Stevin first computed notes 7, 4, and
5, that is, F^, Eb, and E. These involve no more difficult roots
than cubic and quartic. There is now sufficient material to com-
pute the remaining notes by proportion, "the rule of three." Thus
the fifth note (7937), divided by the fourth (8408), gives the second
(5440). This method is much easier than to extract the roots for
each individual note, which runs into difficulties with the roots of
prime powers, as for notes 2, 6, 8, and 12 (C*, F, G, B), where
the 12th root itself must be extracted. But the method by pro-
portion lacks in accuracy, for an error for any note is magnified
in succeeding notes. Even so, the maximum error is only .4
cent. The deviation for Stevin 's monochord lies between those
for Schrbter's two monochords.

Stevin has worked out a second monochord for equal temper-
ament upon the same principle as the first, but with a different
order of notes. 53 Here the maximum error, for E, is 1 cent.
The fact that the two monochords do differ indicates that pro-
portion is not the ideal method (see Table 69).

At the same time that Stevin was setting down the figures for
equal temperament, or perhaps a few years earlier (1595), Prince
Tsai-yii in China was making a much more elaborate and careful
calculation of the same roots of 2.^4 We are not told how he
performed his calculation, but, since it is correct to nine places,
he must have extracted the appropriate root for each note sepa-
rately—and without the aid of logarithms, which were to simplify

Table 69.

Stevin's

Monochord,

No. 2

Lengths

10000

9438

8908

8404

7936

7491

7071

Names

Lengths

6674

6298

5944

5611

5296

5000

Names

53Ibid., p. 72.

54pere Joseph Maria Amiot, De la musique des Chinois (Memoires concernant
l'histoire, . . . des Chinois, " Vol. VI | Paris, 1780]), Part 2, Fig. 18, Plate
21. See also J. Murray Barbour, "A Sixteenth Century Approximation for
IT," American Mathematical Monthly, XL (1933), 69-73.

TUNING AND TEMPERAMENT

the problem so greatly for men who attempted it a few decades
later. In some cases, since the tenth digit will be 5 or larger,
modern computers would round off the number at the ninth digit
by substituting the next higher digit. This is a convention of our
mathematics, intended to reduce the error arising from rounding
off a number. Tsai-yli never did this.

Probably the first printed solution of equal temperament in
numbers was made in Europe in 1630, a generation after Tsai-
yii's time, when Johann Faulhaber solved a problem propounded
by Dr. Johann Melder of Ulm.55 The problem was to divide a
monochord 20000 units in length, so that all intervals of the same
size should be equal. Faulhaber did not explain to his readers
how he had arrived at his result (Table 71), presenting it rather
as a riddle. His monochord was for equal temperament, but con-
tained several errors of 1 in the unit's place. This is the sort
of error likely to occur when logarithms are used, and we might
suppose Faulhaber had made use of the logarithmic tables printed
in his book.

Table 70. Tsai-yii's Monochord

500,000,000

749,153,538

529,731

,547

793,700,525

561,231

,024

840,896,415

594,603,557

890,898,718

629,960,524

943,874,312

667,419,927

1000,000,000

707,106,781

Table 71 .

Faulhaber 's Monochord

Lengths

20000

18877

17817

16817

15874

14982 14141

Names

F x

Lengths

13347

12598

11891

11224

10594

10000

Names

55 Johann George Neidhardt, Sectio canonis harmonici (Ktinigsberg, 1724),
p. 23.

EQUAL TEMPERAMENT

Mersennehas given a number of different tables of equal tem-
perament. The most characteristic, to six places, was furnished
by Beaugrand, "very excellent geometer. "56 Mersenne also
printed a table of first differences for the numbers in this mono-
chord, to be used in connection with a method by Beaugrand for
constructing the equal semitones. A comparison with Tsai-yu's
table shows this one to be very inaccurate, the errors being much
larger than if logarithms had been used.

A much more ambitious table was contributed by Galle.57 In
this table the lengths were given to eleven places. Beside it
Mersenne printed a table with 144,000,000 as fundamental, so
that the numbers might readily be compared with those of "the
perfect clavier with 32 keys or steps to the octave," which had
been presented in the book on the organ. This table will not be
included here, for it seems likely that Mersenne himself com-
puted these numbers from Galleys larger table, by multiplying
them by .00144. Of the numbers in the table, the length for D is
correct to only five places. The others agree fairly well with
Tsai-yii to the ninth place, although there are some slight diver-
gences. Beyond the ninth place no digits are correct. If Galle
was using logarithms, he made some serious errors in interpo-
lation. But if he was extracting roots, it is difficult to see how
he failed to find correctly the middle number, the length for F#,
which represents 1011 times the square root of 1/2. It should
be ten units larger. The length for E*5 (1011 times the fourth
root of 1/2) agrees neither with the correct value nor with the
square root of the length for F#.

Our final table from Mersenne^ Was supplied by Boulliau,
"one of the most excellent astronomers of our age." In it he ex-
pressed the string- lengths for equal temperament in degrees,
minutes, and seconds. This is equivalent to having a fundamental
of 14400 in decimal notation, and the errors should be no greater
than for such a table. However, the errors are greater than in
Stevin's four-place table, with a mean deviation of about 1 cent.
We can only surmise how Boulliau computed his figures. Evi-

^"Mersenne, Harmonie universelle, p. 38.
5*7 Ibid., Nouvelles observations, p. 21.
58Ibid., pp. 384 f.

TUNING AND TEMPERAMENT

dently the sexagesimal notation is somehow linked with his
method of extracting the roots.

Neidhardt printed six-place tables in equal temperament from
Faulhaber, Mersenne, and Biimler, as well as several of his
own. 59 His first original method was to divide the syntonic
comma arithmetically, thus giving rise to a twofold error. The
arithmetical division makes little difference, but the fact that the
syntonic comma is about two cents smaller than the ditonic
comma means that each fifth will be about .2 cent sharper than
in correct equal temperament. Such a division is fairly easy to
make, and, as the cents values indicate, the errors are small.
The mean deviation is about 1 cent.

Later, Neidhardt^O Was to divide the ditonic comma, both
arithmetically and geometrically, the latter method being genuine
equal temperament. He contended, however, that the differences
between these two methods were negligible. Since the greatest
variation is 5 units, in tables containing 6 digits, his contention
was correct. Note that the numbers for the arithmetical division
are the larger throughout the table. The true values come closer
to his geometrical division, but in every instance lie between the
two.

Neidhardt's contemporary, Jakob Georg Meckenheuser,61
printed a table, "as computed in the first Societats-Frucht,"
evidently the proceedings of some learned society. From his
figures, the syntonic comma is divided arithmetically, as in
Neidhardt's first monochord. But evidently Meckenheuser's
division ran to sharps, for seven of his notes were higher in
pitch than the corresponding notes in Neidhardt's monochord.
The higher C is not a true octave, but a B# tempered by a full
syntonic comma, just as his F is really a tempered E#. The
ratio of these pairs of enharmonic notes is the schisma, about
2 cents. Thus even when two temperaments are constructed upon
the same hypothesis and both are intended for equal temperament,

59Neidhardt, Sectio canonis harmonici, p. 32.

SOlbid., p. 19.

"Ipie sogenannte allerneueste musicalische Temperatur (Quedlinburg, 1727),
p. 51.

EQUAL TEMPERAMENT

Table 72. Beaugrand's Monochord

Lengths

200000 188770 178171

168178 158740 149829 141421

Names

C x D

x E F x

Lengths

133480 125992 118920

112245 105945 100000

Names

G x A

x B C

Table 73. Galle

's Monochord

50,000,000,000

F 74,915,353,818

52,973,154,575

E 79,370,052,622

56,123,102,370

x 84,089,641,454

59,460,355,690

D 89,090,418,365

62,996,052,457

x 94,387,431,198

66,741,992,715

C 100,000,000,000

70,710,678,109

Table 74. Boulliau's Monochord

Sexagesimal Notation Decimal Notation The Same, 20000 as Fundament

7200 10000

7632 10600

8092 11239

8573 11907

9072 12600

9605 13340

10179 14138

10772 14961

11405 15840

12110 16819

12823 17810

13580 18861

C 2°

B 2

x 2

A 2

x 2

G 2

x 2

F 2

E 3

x 3

D 3

x 3

C 4 0 0

14400

20000

TUNING AND TEMPERAMENT

Table 75. Neidhardt's Division of Syntonic Comma

Lengths

200000

188867

178148

168229

158683

149845

141344

Names

Cents

99.1

200.3

299.5

400.6

499.9

601.0

Lengths

133472

126041

118888

112268

105898

100000

Names

B*>

Cents

700.2

799.3

900.5

999.7

1100.8

1200

Table 76. Neidhardt's Division of Ditonic Comma

•

Arithmetical

100000

105948

112247

118922

125994

133484

141424

149831

158743

168182

178182

188779

200000

Geometrical
100000
105945
112245
118920
125991
133483
141420
149830
158739
168178
178179
188774
200000

EQUAL TEMPERAMENT

there may be a lack of agreement unless the process is followed
through in exactly the same way for both. If it is true equal tem-
perament, however, it does not matter in what order the notes
are obtained, whether on the sharp or the flat side or mixed up
in anyway whatever. In Table 77, Meckenheuser's numbers have
been divided by 18. This tends to conceal his rather obvious
arithmetical division of the comma: in the original, every num-
ber except one (the length for D) ends in zero. There the value
for G had been 240200000. This has been corrected to 240250000,
since the number should be 240000000 tempered by 1/12 x 1/80 =
1/960.

Since the syntonic comma is much easier to form than the
ditonic, it is easy to see why it should have been preferred as the
quantity to be divided. However, since the ratio of the two commas
is about 11:12, an excellent approximation for equal temperament
can be made by tempering the fifths by 1/11 syntonic comma. 62
This was done arithmetically by Sorge,with the results shown in
Table 78. The mean tempering of his fifths is 1/886, whence the
ratio of the fifth will be .667419962 . . . , instead of .667419927 ....
However, there are larger errors for most notes, since the tem-
perament is not built solely by fifths, and the temperament as a
whole is comparable to Neidhardt's arithmetical division of the
ditonic comma.

Table 77. Meckenheuser's Division of Syntonic Comma

Lengths 200,000,000 188,658,258 178,148,341 168,045,776 158,684,002

Names C C^ D D# E

Cents 0 101.0 200.3 301 3 400.6

Lengths 149,685,380 141,346,458 133,472,222 125,903,184 118,889,159

Names E* F^ G G# A

Cents 501.6 600.9 700.2 801.2 900.5

Lengths 112,147,215 105,899,532 99,894,201

Names A* B B^

Cents 1001.5 1100.8 1201.8

62Marpurg, Versuch iiber die musikalische Temperatur, p. 177.

TUNING AND TEMPERAMENT

Table 78. Sorge's Division of Syntonic Comma

Lengths

200000

188775

178182

168181

158743

149831

141422

Names

Lengths

133484

125994

118923

112247

105948

100000

Names

The impression is likely to become quite strong as one reads
the second half of this chapter that equal temperament is nothing
but a mass of figures of astronomical size. Actually, as far as
the ear is concerned, a wholly satisfactory monochord in equal
temperament (or any other tuning system) would be obtained from
the division of a string a meter long, marked off in millimeters.
Mersenne63 gave such a table, considering it more practicable
than the very complicated tables of Beaugrand and Galle. It could
easily have been constructed from one of the more elaborate
tables by rounding off the numbers at three places. Oddly, many
of Mersenne's figures are one unit too large. The correct mono-
chord is shown in Table 79. It is instructive to note that the de-
viation for this monochord is larger than for one of Marpurg's
irregular tunings, 64 and about the same as that for a couple of
his other tunings. Thus, to three places, Marpurg's systems
would have coincided with equal temperament.

Table 79. Practical Equal Temperament, after Mersenne

Lengths

1000

944

891

841

794

749

707

667

Names

Cents

99.8

199.8

299.8

399.4

500.3

600.3

701.1

Lengths

630

595

561

530

500

Names

Cents

799.9

898.9

1000.7

1099.9

1200

M.D

. .60; S.D. .81

^Harmonie universelle,

p. 339.

^Compare Marpurg's Temperaments E, B, and G in Chapter VH with the
cents values of Table 79.

EQUAL TEMPERAMENT

In 1706 young Neidhardt, full of importance as the author of
a new book on temperament, Beste und leichteste Temperatur
des Monochordi, held a tuning contest with Sebastian Bach's
cousin, Johann Nikolaus Bach, in Jena. 6 5 Neidhardt tuned one
set of pipes byamonochord he had computed by making an arith-
metical division of the syntonic comma. Therefore, although he
had worked out this division to six places, it was about as accurate
as the practical monochord given above. Bach tuned another set
of pipes entirely by ear, and won the contest handily, for a singer
found it easier to sing a chorale in BD minor in Bach's tuning
than in Neidhardt's.

Perhaps part of Neidhardt's difficulty lay in the fact that it is
difficult to tune a pipe to a string. Many years later, Adlung
wrote that this same Johann Nikolaus Bach had what might be
called a "monopipe"— a variable organ pipe with a sliding cyl-
inder upon which the numbers of the monochord were inscribed. 66
Because of the end correction for a pipe, this method is likely to
be faulty. However, forty years before the date of the historic
tuning contest in Jena, Otto Gibelius67 described and pictured
just such a pipe, intended for his meantone approximation dis-
cussed in Chapter m. He also gave an end correction, amount-
ing to 8/3 the width of the mouth of the pipe. In his accurately
drawn copperplate (see Figure F) the width of the mouth is 11
millimeters, making the end correction about 30 millimeters.
Since the internal depth is about 15 millimeters, his rule cor-
responds very closely to our modern rule that the end correction
for a rectangular pipe is twice the internal depth. The Dayton
Miller Collection now at the Library of Congress contains several
specimens of the "tuning pipe," most of them fairly small.

Since the "tuning pipe" was not widely disseminated, organ-

65Philipp Spitta, Johann Sebastian Bach, trans. Clara Bell and J. A. Fuller-
Maitland (2 vols.; London, 1884), I, 137 f.

66Jacob Adlung, Anleitung zu den musikalischen Gelahrtheit (Erfurt, 1758),
p. 311. In addition to the Neidhardt- Bach test, he described a similar ex-
perience that befell Meckenheuser in Riechenberg vor Goslar, where he
tried for three days to tune the organ by his monochord, but in vain. See
Jacob Adlung, Musica mechanica organoedi (Berlin, 1768), p. 56.

67propositiones mathematico-musicae, pp. 1-11.

TUNING AND TEMPERAMENT

bC
c
o
U

(X T3
bD »

c o

C T3
3 O

EQUAL TEMPERAMENT

ists tuning by the aid of the monochord probably had no more
success than Neidhardt had. It is probable, however, that, like
Johann Nikolaus— and Sebastian, too— the organists did not bother
with a monochord but relied upon their ears. Hence the tuning
rules given in the beginning of this chapter were of the greatest
possible importance in practice. Some of them seem so vague
that they would have needed to be supplemented by oral direc-
tions. But if we could be sure that Mersenne's rule that a tem-
pered fifth should beat once per second was to have been applied
to the fifths in the vicinity of middle C, we would have as accurate
a rule for equal temperament as that given by Alexander Ellis
over two centuries later.

Unfortunately, the more mathematically minded writers on
equal temperament have given the impression that extreme ac-
curacy in figures is the all- important thing in equal tempera-
ment, even if it is patent that such accuracy cannot be obtained
upon the longest feasible monochord. This is why Sebastian Bach
and many others did not care for equal temperament. They were
not opposed to the equal tuning itself , and their own tuning results
were undoubtedly comparable to the best tuning accomplished
today— upon the evidence of their compositions, as will be dis-
cussed in the final chapter. But they needed a Mersenne to tell
them that the complicated tables could well have had half their
digits chopped off before using, and that, after all, a person who
tunes accurately by beats gets results that the ear cannot dis-
tinguish from the successive powers of the 12th root of 2.

Chapter V. JUST INTONATION

The seeds of just intonation had been sown early in the Christian
era, when Didymus and Ptolemy presented monochords that con-
tained pure fifths and major thirds (see Chapter II). But they
remained dormant during the Middle Ages. Even after the seeds
had sprouted near the beginning of the modern era, the plants
were to bear fruit only occasionally and haphazardly.

Enough of our metaphor. We shall consider in this chapter
all 12-note systems that contain some arrangement of pure fifths
and major thirds. The Pythagorean tuning may be thought of as
the limiting form of just intonation, since it has a great many
pure fifths, but no pure major thirds. As the various chromatic
notes were added to the scale during the latter Middle Ages, they
were tuned by pure fifths or fourths to notes already present in
the scale. Finally, fifteenth century writers were describing
the formation of a complete chromatic monochord, using the
Pythagorean intervals. Such a writer was Hugo de Reutlingen,
whose altered notes consisted of two sharps and three flats.
Since the more typical tuning has G# instead of AD, that is shown
in Table 80. Of course the deviation would be the same as for
Hugo's tuning. The ratio for each diatonic semitone is 256:243,
and for the chromatic semitone 2187:2048. Compare with these
ratios the relative simplicity of the ratios for Marpurg's first
tuning, the model form of just intonation. (The lengths are very
much simpler also.)

The first known European writer to break away from the
Pythagorean tuning for the tuning of the chromatic monochord
was Bartolomeus Ramis de Pareja.2 Ramisgave specific direc-
tions for tuning the monochord that resulted in a system in which
the six notes A^° -G° are joined by perfect fifths, as in the Pyth-
agorean tuning, and the remaining six notes, D^-F*-1 , also
joined by fifths, lie a comma higher than the corresponding notes

^Flores musicae omnis cantus Gregoriani (Strassburg, 1488), Chapter EL

^Musica practica (Bologna, 1482); new edition, by Johannes Wolf (1901), pub-
lished as Beiheft der Internationale Musikgesellschaft.

TUNING AND TEMPERAMENT

Table 80. Pythagorean Tuning

Lengths

629856

589824

559872

531441

497664

472392

442368

Names

C°

C^°

D°

EbP

E°

F°

F*o

Cents

114

204

294

408

498

612

Lengths

419904

393216

373248

354294

321776

314928

Names

G°

G#o

Bbo

B°

C°

Cents

702

816

906
M.D. 11.7;

996
S.D. 11.8

1110

1200

Names C
Cents 0

C* D

Table 81. Ramis' Monochord

bO -1 0 -W~1_0 h°

92 182 294 386 498 590 702 792 884 996 1088 1200
M.D. 10.0; S.D. 10.1

in the Pythagorean tuning (see Table 81). Thus there are pure
major thirds to only the four notes BD-G.

Montucla,^ writing a "history of music," gave string-lengths
for a 17 -note tuning, in which twelve notes are the same as in
Ramis. The other five extend the scale to A*-1 and to GD'°.
This is a wholly useless extension because such enharmonic pairs
as D^0 and C*"1 differ by the schisma, 2 cents. Helmholtz
was more astute in constructing his 24 -note harmonium in just
intonation, in which the eight notes from C° through C#° are joined
by fifths; the next eight, E"1 through E#_1 , furnish major thirds
to notes in the first series; and the remaining eight, AD+1 through
A+1, are considered (by disregarding the schisma) as equivalent
to the thirds above the notes in the second series, i. e., G#~2-
Gx"2.4

Ramis' monochord does not differ perceptibly from the Pyth-
agorean tuning. If he had substituted DD° and all the other Pyth-
agorean enharmonic equivalents of the syntonic notes, he would

3jean Etlenne Montucla, Histoire des mathe'matiques (New ed.; Paris, 1802),
IV, 650.

4H. L. F. Helmholtz, Sensations of Tone, pp. 316 f.
90

JUST INTONATION

have had a monochord from E^b0 through G°, in Pythagorean
tuning. His reason for making the new division was solely to
simplify the construction of the monochord. In his own words,
the Pythagorean tuning, as given by Boethius, is "useful and
pleasing for theorists, but tiresome for singers and irksome to
the mind. But because we have promised to satisfy both [singers
and theorists], we shall simplify the division of the monochord."
Later he expressed the same idea in these words: "So therefore
we have made all our divisions very easy, because the fractions
are common and are not difficult."

Undoubtedly Ramis' method is easier. But if he had desired
to obtain the equivalent of the Pythagorean tuning from AD to C#,
he would have commenced his tuning with F# instead of with C,
having notes with zero exponents from D° to C*° and with -1
from G#-1 to Fx-1 . On such a monochord, however, as on the
usual Pythagorean monochord, the eight most common thirds
would have been very sharp and the four useless thirds, E-A",
B-E13, F^-B^, and C*-F, would have been pure. The monochord,
as Ramis actually tuned it, has as its four pure thirds, B^-D,
F-A, C-E, and G-B. Thus, although Ramis professed to be mak-
ing his division of the octave solely for the sake of simplicity,
the accidental result was that several pure triads were available
in keys frequently used.

The bitter critics of Ramis in his own day failed to realize
that his tuning was just what he had described: a simplified
equivalent of the Pythagorean tuning — shifted, however, by six
scale degrees to the flat side. To them, any tampering with the
old intervals was sacrilege. Many later writers, misled by
Ramis' announced intentions, have stated, without examining his
monochord, that he had advocated temperament. As we have de-
fined temperament and as the word is usually understood, this is
a serious misconception. It has even been stated that Ramis ad-
vocated equal temperament! Since Ramis' book is accessible in
a modern edition, there is no longer any excuse for repeating
such myths.

It must be said, somewhat sadly, that Ramis was not aware
himself of the peculiar properties of the monochord he had

TUNING AND TEMPERAMENT

fathered. For example, he explained that although Eb does not
form a major third to B, D# is not really needed, for the minor
triad B D F* can be used in making a Phrygian cadence on E.
But his interval B^-E^0 is slightly better than the Pythagorean
thirds, Ab°-C° and Eb°-G°, that were acceptable to him!

Ramis must have been a good practical musician. Although
his system would not now be called a temperament, we might do
well to take him at his own evaluation and hail him as the first
of modern tuning reformers.

Corroboration of Ramis' tuning system is found in an inter-
esting anonymous German manuscript of the second half of the
fifteenth century, Pro clavichordiisfaciendis, which Dupont^ ran
across in the Erlangen University Library. Starting with the
note B, C is to be a just semitone (16:15) higher, E a perfect
fourth, G a just minor sixth (8:5), etc. A succession of pure
fifths on the flat side extends to Gb, below which there is a just
major third (5:4), E^b, and the monochord is completed by add-
ing BbD, the fifth above Ebb! The complete monochord is shown
in Table 82.

The deviation for this tuning is almost precisely the same as
for that of Ramis, and it too contains many pure fifths and sev-
eral pure thirds. However, it has one peculiar feature as Du-
pont has presented it. In every other tuning system we have ex-
amined, there has been an uninterrupted succession of notes
connected by fifths from the flattest to the sharpest. In the Pyth-
agorean and other regular tuning systems, such as the meantone,
the wolf fifth would be very flat or sharp, and in the irregular
systems there would be other divergences. But the note names
persisted, usually from Eb to G* inclusive.

Table 82. The Erlangen Monochord

Names C° Db° Ebb+1Ebo E"1 F° Gb° G° Ab° Bbb+1Bb° B_1 C°
Cents 0 90 202 294 386 498 588 702 792 904 996 1088 1200

M.D. 10.3; S.D. 10.5

^Wilhelm Dupont, Geshichte der musicalischen Temperatur (Erlangen, 1935),
pp. 20-22.

JUST INTONATION

But in the Erlangen monochord there is no D or A, and the
notes that Dupont has given as their enharmonic equivalents,
E^b and B^b, are not in a fifth -relation with any other notes in
the monochord. Therefore it seems obvious that the anonymous
writer intended these notes to be D° and A0, each of which is
higher by the schisma than E^b*1 and Bbb+1 respectively. Then
the notes that are pure thirds above D° and A0 will be F*-1 and
C^"1 , notes that continue the fifth-series from B"1. It would
then be immaterial whether to call the semitone between G and
A by the name AD° or G* * , since either would complete the
scale correctly. The original writer, by the way, had not named
the black keys, merely designating the semitone between C and
D as the first, between D and E as the second, between F and G
as the third, and between G and A as the fourth. In renaming
some of the black keys, therefore, we are not violating his in-
tent, but rather confirming it. The revised monochord, with
schismatic alterations, is shown in Table 83.

These two pre-sixteenth-century tunings, the one in Spain
and the other in Germany, are sufficient indication of the trend
of men's thinking with regard to consonant thirds. Lodovico
Fogliano," half a century later than Ramis, offered no apologies
for using the 5:4 ratio for the major third. But he was not con-
tent to present ordinary just intonation. Realizing that D° formed
an imperfect fifth below A"1 , he advocated D-1 as a consonant
fifth. This in turn led him to BD° as a pure major third below
D-1, as well as the Bb+1 as third below D°. But he said the
"practical musicians" used only one key each for D and B*3,
"neither right nor left, but the mean between both." "Such a
mean D or BD, moreover, is nothing else than a point dividing
the proportion of the comma into two halves."

Table 83. Erlangen Monochord, Revised

Names C° C*"1 D° E* E_1 F° F^G0 G*" A° Bb° B_1 C°
Cents 0 92 204 294 386 498 590 702 794 906 996 1088 1200

M.D. 10.0; S.D. 10.1

6Musica theorica (Venice, 1529), fol. 36.

TUNING AND TEMPERAMENT

To obtain the mean proportional by geometry, Fogliano used the
familiar Euclidean construction, and appended a figure to show how
the division was to be made. This alteration of pure values, he said
is "what they [the practical musicians] call temperament." Here
is the germ of the meantone temperament, which his countryman
Aron had described in its complete format aboutthis same time.

For the sake of showing monochords in just intonation from
the early sixteenth century, there are set down here three mon-
ochords after Fogliano, first with his one pair of D's and B^'s,
then with the second pair, and finally with the mean D and B*3.
The first monochord (Table 84) is the best, having two groups of
four notes each with like exponents. The second monochord
(Table 85) would have had the same deviation as the first if it had
had F#_1 (in place of F#"2) as third above D°. (This is Mar-
purg's first monochord, Table 96.) The monochord with the two
meantones (Table 86) ranks between the first two. If Fogliano
had formed three meantones, including one on F#, the deviation
would be slightly less than for the first monochord. The result
is given in Table 87.

Table 84. Fogliano's Monochord, No. 1

Lengths

3600

3456

3240

3000

2880

2700

2592

2400

Names

C°

c*-2

D"1

Eb+i

E"1

F°

F#-2

G°

Cents

182

316

386

498

568

702

Lengths

2304

2160

2025

1920

1800

Names

G,_2

A"1

Bbo

B_I

C°

Cents

772

884

996

1088

1200

M.D. 21.3; S.D. 23.6

JUST INTONATION

Table 85. Fogliano's Monochord, No. 2

Lengths

3600

3456

3200

3000

2880

2700

2592

2400

Names

C°

c#-2

D°

Eb+1

E"1

F°

F*-2

G°

Cents

204

316

386

498

568

702

Lengths

2304

2160

2000

1920

1800

Names

Gfr.

A-1

Bb+i

B"1

C°

Cents

772

884

1018

1088

1200

M.D. 25.0; S.D. 26.7

Table 86. Fogliano's Tempered Just Intonation

Lengths

3600

3456

|_3220j

3000

2880

2700

2592

2400

Names

C°

c*-2

D-i

Eb+i

E"1

F°

F*«

G°

Cents

193

316

386

498

568

702

Lengths

2304

2160

[2012.5]

1920

1800

Names

G#-2

A"1

Bb+I

B"1

C°

Cents

772

884

1007

1088

1200

M.D. 23.2; S.D. 24.7
Table 87. Fogliano's Tempered Just Intonation, Revised

Lengths

3600

3456

[3220]

3000

2880

2700

[2576]

Names

C°

c#-2

1
D 2

Eb+1

E"1

F°

F# 2

Cents

193

316

386

498

579

Lengths

2400

2304

2160

[2012.5]

1920

1800

Names

G°

G#-l

A'1

Bb+i

B"1

C°

Cents

702

772

884
M.D. 21.3;

1007
S.D. 22.3

1088

1200

TUNING AND TEMPERAMENT

Martin Agricola' resembled Ramis in his tuning ideas. He
gave a monochord in which the eight diatonic notes, including B*3,
were joined by pure fifths, as in the Pythagorean tuning. Then
he directed that the interval from B to the end of the string be
divided into ten parts, with C* at the first point of division, D#
at the second, and G* at the fourth. Then F* was to be a pure
fourth to C*. Thus these black keys were given syntonic values,
and the whole monochord is made up of notes with 0 and -1 ex-
ponents (see Table 88). Ramis' monochord is slightly better than
Agricola' s, with a ratio of 6:6 for the number of fifths in each
group, in place of 8:4.

Table 88. Agricola 's Monochord

Names C° C^D0 D*"1 E° F° F*"1 G° G*-1 A0 Bb° B° C°
Cents 0 92 204 296 408 498 590 702 794 906 996 1110 1200

M.D. 10.3; S.D. 10.5

It will be observed that the better of Fogliano's untempered
monochords has more than twice the deviation of Ramis'. Thus
it might be thought that Fogliano had been unfortunate in his
choice of intervals. Quite the contrary. The most symmetric
form of just intonation for the series ED-G* has four notes with
the same exponent, followed by four more with exponents that
are one less. Of the remaining four notes, two would have +2
and two would have -2 as exponents. This is precisely Fogliano's
second monochord, if we should substitute F^"1 in it. Fogliano's
first monochord has the exponential pattern 1,4,4,3, which is just
as satisfactory. (That is, the tuning contains one note with ex-
ponent + 1, 4 with 0 and -1 exponents, and 3 with -2.) The diffi-
culty, therefore, is inherent in just intonation itself, as will be
discussed further a bit later.

Salomon de Caus^ was one of several mathematicians of the
early seventeenth century who were interested in just intonation.

'"De monochorea' dimensione," in Rudimenta musices (Wittemberg, 1539)

°Les raisons des forces mouvantes avecdiverses machines (Francfort, 1615)
Book 3, Problem III.

JUST INTONATION

If we follow his directions, we obtain the monochord shown in
Table 89. Here there are three groups of four notes each with
the same exponent — the most symmetric arrangement of all.
The deviation is appreciably less than in Fogliano' s arrangement.
Johannes Kepler^ gave some genuine tuning lore together
with an elaborate discussion of the harmony of the spheres. His
two monochords in just intonation (Tables 90 and 91) are identi-
cal except that the second has a G* in place ofanAD. Since Kep-
ler had five notes with zero exponents in both monochords, the
deviation for his systems is lower than most that have been pre-
sented in this chapter.

Table 89. De Caus's Monochord

Names C° C^D"1 D#_2 E"

?#-2 no

1#"« A-l

B_1

Cents 0 70 182 274 386 498 568 702 772 884 996 1088 1200

M.D. 17.7: S.D. 20.1

Table 90. Kepler's Monochord, No. 1

Lengths

1620

1536

1440

1350

1296

1215

1152

Names

C°

c"-»

D°

Eb+i

E_1

F°

jr#-i

Cents

204

316

386

498

590

Lengths

1080

1024

960

900

864

810

Names

G°

G*+i

Bb+i

B"1

C°

Cents

702

794,

906

1018

1088

1200

M.D. 14.0; S.D. 15.8

^Harmonices mundi, p. 163.

TUNING AND TEMPERAMENT

Although Marin Mersenne was a zealous advocate of equal
temperament in practice, he took pains to present literally doz-
ens of tables in just intonation. He repeated, among others,
Kepler's two monochords shown in Tables 90 and 91, together
with tables for keyboards with split keys. Four of his monochords
(Tables 92-95) are worth including here, as evidence of the va-
riety that is possible in a type of tuning that is ordinarily thought
to be fixed and uniform.^ None is as good as either of Kep-
ler's two.

Table 91. Kepler's Monochord, No. 2

Lengths

100000

93750

88889

833333

80000

75000

71111

Names

C°

c#-i

D°

Eb+i

E_1

F°

f'-'

Cents

204

316

386

498

590

Lengths

66667

62500

60000

56250

53333

50000

Names

G°

Ab+i

Bb+i

B"1

C°

Cents

702

814

906
M.D. 14.0;

1018
S.D. 15.8

1088

1200

Table 92. Mersenne's Spinet Tuning, No. 1

Lengths

3600

3375

3240

3000

2880

2700

Names

C°

Db+1

D"1

Eb+1

E_1

F°

Cents

112

182

316

386

498

Lengths

2400

2250

2160

2025

1920

1800

Names

G°

Ab+1

A-1

Bbo

B*1

C°

Cents

702

814

884
M.D. 17.7

996
S.D.

1088
20.1

1200

lOMersenne, Harmonie universelle, pp. 54, 117 f.
98

2531 1/4
nb+i

610

JUST INTONATION

Table 93. Mersenne's Spinet Tuning, No. 2

Lengths

3600

3456

3200

3072

2880

2700

2592

2400

Names

C°

C#-2

D°

D*-2

E"1

F°

F#"2

G°

Cents

204

274

386

498

568

702

Lengths

2304

2160

2025

1920

1800

Names

Gf-2

A"1

B"1

Cents

772

884

996

1088

1200

M.D. 21.3; S.D. 23.6

Table 94. Mersenne's Lute Tuning, No. 1

Names C° Db+1 D"1 Eb+1 E"1 F° Gb+1 G° Ab+1A_1 Bb+1 B"1 C°
Cents 0 112 182 316 386 498 610 702 814 884 1018 1088 1200

M.D. 21.3; S.D. 23.6

Table 95. Mersenne's Lute Tuning, No. 2

Names C° Db+1 D° Eb+1 E"1 F° Gb+1 G° Ab+1 A"1 Bb+1 B_1 C°
Cents 0 112 204 316 386 498 610 702 814 884 1018 1088 1200

M.D. 17.7; S.D. 20.1

Table 96. Marpurg's Monochord, No. 1

Lengths

900

864

800

750

720

675

640

Ratios

24/25

25/27

15/16

24/25

15/16

128/135

15/16

Names

C°

c*-2

D°

Eb+1

E"1

F°

F*'1

Cents

204

316

386

498

590

Lengths

600

576

540

500

480

450

Ratios

24/25

15/16

25/27

24/25

15/16

Names

G°

G*-2

A"1

B"1

C°

Cents

702

772

884

1018

1088

1200

M.D. 21.3;

S.D. 23.6

TUNING AND TEMPERAMENT

Table 97. Marpurg's Monochord, No. 3

Names C° C#_2D° Eb * E^1 F° F#_1 G° G^2 A0 Bb° B"1 C°
Cents 0 70 204 306 386 498 590 702 772 906 996 1088 1200

MD. 19.3; S.D. 22.0

Table 98. Marpurg's Monochord, No. 4

Names C° C^D-1 Eb+1 E _1 F° F#~2G0 G^A"1 Bb +1 B_1 C°
Cents 0 70 182 316 386 498 568 702 772 884 1018 1088 1200

M.D. 25.0; S.D. 26.7

Note that Mersenne's first spinet tuning (Table 92) has flats
for its black keys and the second tuning (Table 93) has sharps
except for BO. The first tuning is constructed exactly the same
as deCaus's tuning (Table 89), except that it begins a major third
lower, with Gb instead of B°. Mersenne's first lute tuning (Ta-
ble 94) differs from his first spinet tuning (Table 92) at only one
pitch (BD+1 instead of B130), but that is enough to increase its de-
viation to that of the second spinet tuning (Table 93). The second
lute tuning (Table 95), although differing from the first spinet
tuning (Table 92) at two places, has the same deviation.

Friedrich Wilhelm Marpurg, *■ who wrote brilliantly about
temperament 140 years after Mersenne, included four mono-
chords in just intonation. The second of these was Kepler's
first, and need not be repeated here. The other three are shown
in Tables 96-98. In each of them the notes, according to their
exponents, are grouped into four classes. The first may be con-
sidered the model form of just intonation, the ideal form of Fog-
liano's second monochord (Table 85).

Opelt has shown two monochords in just intonation from Rous-
seau's Dictionary. *•* The first (Table 99) was by Alexander
Malcolm, whose linear improvement upon just intonation is to be
found in Chapter VII. This is the same as Kepler's second mon-
ochord (Table 91), transposed a fifth lower.

11Versuch liber die musikalische Temperatur, pp. 118, 123.

12F

100

12F. W. Opelt, Allgemeine Theorie der Musik (Leipzig, 1852), p. 46.

JUST INTONATION

Rousseau tried to "improve" upon this tuning by substituting
other just pitches in place of D*3"1"1, F*"1 , and B"0, with very un-
satisfactory results, since his division of the major tone of 204
cents was into semitones of 70 and 134 cents! This monochord
(Table 100) is the reverse of Marpurg's fourth (Table 98), with
semitones paired in contrary motion, when Rousseau's A*3*1 is
made to coincide with Marpurg's G*~2.

Table 99. Malcolm's Monochord

Names C° Db+1 D ° Eb+1 E"1 F° f""1 G° Ab+1 A"1 Bb° B"1 C°
Cents 0 112 204 316 386 498 590 702 814 884 996 1088 1200

M.D. 14.0; S.D. 15.8

Table 100. Rousseau's Monochord

Names Cu C#"2 D° Eb+1 E"1 F° F^"2 G° Ab+l A"1 Bb+1 B"1 C°
Cents 0 70 204 316 386 498 568 702 814 884 954 1088 1200

M.D. 25.0; S.D. 26.7

Table 101. Euler's Monochord

Names C° C^2 D° D#_2 E_1 F° F*"1 G° G*~2A_1 A*"2 B_1 C°
Cents 0 70 204 274 386 498 590 702 772 884 976 1088 1200

M.D. 17.1; S.D. 20.1

Table 102. Montvallon's Monochord

Names C° C#_1 D° E^1 E"1 F° F#_1 G° G#_1 A"1 Bb° B"1 C°
Cents 0 92 204 316 386 498 590 702 794 884 996 1088 1200

M.D. 12.0; S.D. 13.3

101

TUNING AND TEMPERAMENT

Table 103. Romieu's Mono-chord

Names C° C#"2D° Eb+1 E"1 F° F^G0 G^A"1 B°° B"1 C°
Cents 0 70 204 316 386 498 590 702 772 884 996 1088 1200

M.D. 17.7; S.D. 20.1

Euler's monochord ran entirely to sharps. ** However, it has
the same symmetric grouping of its notes as de Caus's (Table
89), only transposed a fifth higher.

Montvallon's monochord, given by Romieu,14 follows a more
familiar order in the selection of notes than Euler's did (see Ta-
ble 102).

Romieu himself contributed an example (Table 103) of a
"syst^me juste." 15 it has a somewhat more complicated pattern
than Euler's (Table 101), but the same deviation.

Theory of Just Intonation

In the foregoing pages there have been presented more than
twenty different monochords in authentic just intonation, i. e.,
with pure fifths and major thirds. Their mean deviations have
varied from 10.0 to 25.0. And yet each has a right to be called
just intonation! This great divergence can be explained by math-
ematics. Let us consider first a monochord in the Pythagorean
tuning. Its mean deviation is 11.7. A Pythagorean chromatic
semitone, as C°-C*°, is 114 cents; the diatonic semitone, as
C#° -D° , 90. Hence the deviation for the pair of semitones is 24
cents. When the just semitones are used, the chromatic semi-
tone, C-C*"1, is 92 cents; the diatonic, C#_1-D°, 112. The
deviation for the pair of just semitones is 20 cents, or 4 cents
less than for the pair of Pythagorean semitones. Therefore the
substitution of each just note reduces the deviation by 4/12 or
.3 cent.

**A. F. H'aser, "Uber wissenschaftliche Begriindung der Musik durch
Akustik," Allgemeine musikalische Zeitung, 1829, col. 145.

14"Me"moire theorique & pratique sur les systemes tempe're's de musique,"
Memoires de l'acade'mie royale des sciences, 1758, p. 867.

15Ibid., p. 865.

102

JUST INTONATION

But the sixth note to be altered around the circle of fifths is
adjacent to the first note to have been altered, and therefore the
total deviation is unchanged. The same is true for the seventh
note. The eighth note lies between two notes, each sharper by
the syntonic comma. Therefore, when it too is raised, the syn-
tonic semitones already present are changed to Pythagorean
semitones, and the deviation is increased by .3 cent. This proc-
ess continues until all twelve notes have been raised by a comma,
and the monochord is again in Pythagorean tuning. If we call the
number of notes with -1 exponent nx , and with 0 exponent n2 , the
following formula gives the mean deviation:

3D, = 29 + n , - 6 +

6 - n, - 6

The minimum deviation of 10.0 cents occurs when (nx ,n2) = (5,7),
(6,6), or (7,5). Thus Ramis' monochord (Table 81) with 6,6 is
one of the three best possible.

When there are notes with three different exponents, the
change of a single note may cause a greater change in the devi-
ation than was possible with two exponents only. Suppose a mon-
ochord contains the notes C° C#_1 D"1, the total deviation being
18 cents for the two semitones. When C* "2 is used, the devia-
tion becomes 42 cents, an increase of 24 cents. But if the notes
had originally been C C#_1 D°, the change to C*"2 would in-
crease the deviation from 20 cents to 64 cents, that is, by 44
cents, or two commas. Again, the deviation of the two semitones
C#_1 D° Eb+1 is 24 cents; with D'1 it is 44 cents, an increase
of 20 cents.

Thus when a note is changed by a comma, the change in the
mean deviation may be 1/3 (as before) or 6/3 or 11/3 or 5/3 . A
much more complicated formula, therefore, is needed to express
the deviation with the three exponents. If we call the number of
notes with -1 exponent nx , with 0, n2 , and with +1, n3 , the mean
deviation is given by the formula:

103

TUNING AND TEMPERAMENT

3 D3 = 23 + ru - 6 + n3 - 6 +

6 - rii - 6

6 - n3 - 6

7(k2 - kx) + 5(k4 - k3), where kx = the larger of n2 and (7 - nx),
k2 = the smaller of 7 and (12 - ni), k3 = the larger of n2 and
(5 - ni), and k* = the smaller of 5 and (12 - m). The terms con-
taining the k's are zero whenever k2 <ki and ki ^k3.

Let us now compute the deviations for two of the tunings
shown on previous pages. Mersenne's first spinet tuning (Table
92) has for its (n1,n2,n3) the numbers (4,4,4). Here ki = n2 = 4,
Ko — • y Ko — n 2 — +■ j ■K-4 — ** •

3 D3 = 23 + 2 + 2 + 0 + 0 + 7x3 + 5x1 = 53. D3 = 17.7.
For Mersenne's second spinet (Table 93) or first lute tuning
(Table 94) the exponential numbers are (4,3,5).

3 D3= 23 + 2 + 1 + 0 + 0 + 7x4 + 5x2 + 64. D3= 21.3.

When there are four different exponents, there is a very an-
alogous formula for the deviation:

3 D4 = 23 - nx + \n1 + n2 - 6 |+|n4 - 6 | +

6 - n. -6

6 -In! + n2 - 6

+ 7(k2 - kx) + 5(k4 - k3) + 7(L2 - LJ + 5(L4 - L3 ),

where kx = the larger of n3 and (7 - nx - n2), k2 = the smaller of
7 and (12 - nx - n^, k3 = the larger of n3 and (5 - nx - n2), k4 =
the smaller of 5 and (12 - nx - n2); L,= the larger of n2 and
(7 - nx), L2 = the smaller of 7 and (12 - nj, L3 = the larger of
n2 and (5 - nt), and L4 = the smaller of 5 and (12 - nj. The
terms containing the k's and L's are zero whenever k2 < kx,
k4 < k3, L 2 < Li, and L4 < L3 .

As examples, let us compute the deviation for two of Mar-
purg's tunings. His first tuning (Table 96) is the model form of
just intonation, with (2,4,4,2) for its (n1,n2,n3,n4). Here kx = 4,
k2 = 6, k3 = 4, k4 = 5, Li = 5, L2 = 7, L3 = 4, and L4 = 5. Hence
3 D4= 23 -2 + 0 + 4 + 1 + 0 + 7x2 + 5x1 + 7x2 + 5x1 = 64.
D4 = 21.3. Marpurg's third tuning (Table 97) has for its expo-

nents (2,3,6,1). Here k2 = 6, k3 = 7,

The deviation:

i - 6, k4 - 5, Lx - 5,

L2 = 7, L3 =4, L4 = 5.

k,
3D

= 23-2 + 1 + 5 + 0

104

JUST INTONATION

+ 0 + 7x1 + 0 + 7x2 + 5x2 = 58. D4= 19.3.

With all these complex mathematical formulas before us, we
are likely to forget that we are ostensibly studying a form of
tuning that to many people is a sort of ideal system. It is not
likely that any sane person would advocate so perverted a tuning
as that represented by (5,1,1,5), with a mean deviation of 43.3
cents. But the systems that have been shown on the previous
pages have all been advocated by various writers, and they show
great variety in their construction and almost as great a variety
in their deviations, ranging from the 10.0 of Ramis to the 25.0
of Fogliano's second or Rousseau's or Marpurg's fourth. The
model form, Marpurg's first, with a deviation of 21.3, comes
nearer the maximum than the minimum. We shall speak again
of just intonation in the final chapter. Let us close this chapter
with a double paradox: there is no such thing as just intonation,
but, rather, many different just intonations; of these, the best is
that which comes closest to the Pythagorean tuning.

105

Chapter VI. MULTIPLE DIVISION

If a keyboard instrument is not in equal temperament, its intona-
tion can be improved by a judicious increase in the number of
notes in the octave. The first reference to split keys came from
Italy, where before 1484 the organ of St. Martin's at Lucca had
separate keys for E^ and D* and also for G* and A*3.* At this
same time, Ramis^ noted that split keys were being used in Spain,
but objected to having separate keys for A*3 and G* and for F*
and G*3, on the ground that this would be mixing the chromatic
with the diatonic genus. From Germany came further evidence
of the divided keyboard from Arnold Schlick,^ who referred to
an organ constructed at the turn of the sixteenth century "that
had double semitones on manual and pedal . . . which were called
half semitones or 'ignoten.'"

There are frequent references to multiple division during the
sixteenth and seventeenth centuries, chiefly by Italian theorists.
Jean Rousseau^ in 1687 deplored the fact that the French clave-
cins did not have the "doubles feintes" common inltaly, and con-
sequently had "mauvais effets dans les Tons transposez." But
the split keys must have been very common in Germany during
the latter part of the seventeenth and beginning of the eighteenth
centuries, if we may judge by the copious references to "sub-
semitonia" by Werckmeister and his successors. Buttstett, it is
true, said in 1733 that the sub- and supersemitonia were "mehr
curieux als practicabel."5 But six years later, in Holland, van
Blankenburg was to show u't Gesnede Clavier" with three extra

Iwilhelm Dupont, Geschichte der musicalischen Temperatur, p. 45.

^Musica practica, Tract. 2, Cap. 4.

^Spiegel der Orgelmacher und Organisten (Maintz, 1511), Chap. 8. Reprinted in
Monatshefte fiir Musikgeschichte, 1869.

4Traite~ de la viole (Paris, 1687), p. 50.

^Johann Heinrich Buttstett, Kurze Anfiihrung zum General-Bass (2nd edition;
Leipzig, 1733), p. 20.

TUNING AND TEMPERAMENT

keys, as well as an "Archicymbalam" with eighteen notes in the
octave. 6

Handel played on English organs with split keys.' Father
Smith's Temple Church organ in London, constructed in 1682-83,
had the same pairs of divided keys as the Lucca organ, G#-Ab
and D#-Eb, and so did Durham Cathedral. The organ of the
Foundling Hospital (1759) had an ingenious mechanism by which
Db and Ab could be substituted for C# and D#, or D# and A# for
Eb and Bb, thus increasingthe compass to sixteen notes, without
increasing the number of keys.

Many of the sources said nothing about the tuning of the extra
notes, and we can freely assume that whatever variety of mean-
tone temperament was used for the twelve regular notes was ex-
tended both clockwise and counterclockwise around the circle
(or, rather, spiral) of fifths. More interesting to us are the sys-
tems that represent just intonation, as extended to the enharmonic
scale. We have already noted that Fogliano (1529) had felt the
need for two D's and two Bb's, to ensure just triads, but was
willing to settle for a mean D and a mean B*3. But van Blanken-
burg, mentioned above, included both pairs of notes inhisArchi-
cymbalam, and so did almost all of the men whose systems will
be described below.

The "enharmonic genus" of Salinas" was one of the earliest
and best of these systems. Although it contained twenty-four
notes, it had nothing in common with a real enharmonic scale
composed of quarter tones. It is just intonation extended to seven
sharps and six flats. In tabular form it would appear as shown
in Table 104.

Observe that all the notes in the right diagonal are duplicated
on the left, a comma lower. Thus it is possible to play all major
triads from Gb through G*, and all minor triads from Eb through
E*. Mersenne's "parfait diapason" ^ is based upon Salinas' sys-
tem, with the addition of seven more notes, or thirty-one in all

"Quirinus van Blankenburg, Elementa musica (The Hague, 1739), p. 112.
'Helmholtz, Sensations of Tone, p. 434.
8De musica libri VII, p. 122.
^Harmonie universelle, p. 338.

108

MULTIPLE DIVISION

(see Figure G). These would be joined to Table 104 on the left
side, as shown in Table 105.

Table 104. Salinas' Enharmonic Genus

A#-2 E#-2 B#-2

,#-i n¥-i n#-i

»#-!

yfl-1 Cf-1 Qff-1 DP-1 A

D° A° E° B° F*°

Bb+i F+i c+i G+i D+1

Gb+2 Db+2 Ab+2 Eb+2 Bb+2

Gb+3

f-1

Fig. G. Mersenne's Keyboard with Thirty-One Notes in the Octave
(From Mersenne's Harmonie universelle)

Reproduced by courtesy of the Library of Congress

Table 105. Mersenne's Addition to Salinas' System
A"

E"1 B"1
G°

Ab+i Eb+i

This is not a particularly clever addition. Note that Mersenne
did not have a C°. Furthermore, for the sake of symmetry,
there should have been Db+1 in the lowest line of Mersenne's
additional notes, Bbb+2 , Fb+2 , and Cb+2 in the line below it, and
C*~2, G*"2, and D*~2 in the line above the highest line, or a total
of thirty-nine notes.

109

TUNING AND TEMPERAMENT

The praiseworthy thing about Mersenne's addition is that it
recognized the need for having more pairs of notes differing by
a comma. Imperfect as his scheme was, it would be much more
useful than the 34-note keyboard of Galeazzo Sabbatini, given by
Kircher. 10 There were, as usual with Kircher, many errors in
the figures, and an erratic manner of naming the notes. The ac-
tual notes of Sabbatini' s keyboard are shown in Table 106.

Table 106. Sabbatini's Keyboard

Cx-3 Gx-3 Dx-3 Ax

A*"2

E#_2 B#-2 [F#-2]

F#-i

C*"1 G*-1 D#_1

D°

E° Bu

Bb*

F+1

C+1 G+1

|Gb+2] Db+2

Ab+2

Eb+2

3 Bbb+3

Fb+3

cb+3

gbb+3

cbb^ Gbb^ Dbb^ Abb

Ebbb+5 Bbbb+5 [pbb+5]
Dbbb-^

Except for the three notes in brackets which have been sup-
plied, this is a beautifully symmetric scheme. But how different
from that of Salinas! Here there are no notes differing by the
syntonic comma, with the result that no major triad based on a
note in the diagonal on the right will have a pure fifth, and there
will be a similar series of defective minor triads. With this in-
tonation it is not even possible to supply a missing note by its
enharmonic equivalent, because no pair of notes differs by the
ditonic comma either. The most characteristic small interval
in it is the great diesis of 42 cents, as between A#~2 and BD+1,
whereas A^"1, needed as the fifth of the D* triad, lies almost
half way between these two notes, 22 cents higher than A*"2 and
20 cents lower than B^*1. Other small intervals of little use
contain 28, 14, and 8 cents. This, then, is an example of just

l^Athanasius Kircher, Musurgia universalis, I, 460.
110

MULTIPLE DIVISION

intonation carried to an absurd end.

Doni's three-manual organ keyboard* * (abacus Triharmon-
icus) was more elaborate than any system previously described,
with sixty keys in the octave, but with only thirty -nine distinct
pitches. The lowest keyboard was the Dorian, then the Phrygian,
and finally the Lydian. The arrangement of the notes on each
keyboard was identical, and the keyboards were tuned a major
third apart, so that the Dorian E, the Phrygian C, and the Lydian
A^ were the same pitch. The tuning was largely just, as can be
seen from Table 107, which represents seventeen of the twenty
notes on one keyboard.

Table 107. Doni's Keyboard

2 JT#-2 C#-2 Q#-2 D^2

D-1 A"1 E"1 B"1

*b°

C° G°

Gb+i Ab+i Eb+i

This arrangement is somewhat lacking in symmetry, and the
additional three notes, which were real quarter tones, were of no
use except to illustrate the scales of the Greeks, this being one
of the uses of the organ. The enharmonic notes were formed, as
Didymus formed his, by an arithmetical division of the syntonic
semitone, 16:15, into 32:31 and 31:30 quarter tones. 2

The nineteenth century was particularly rife with proposals
to increase greatly the number of notes in the octave. Many of
the instruments upon which the inventors practiced their ingen-
uity were harmoniums, intended for experimental purposes only.
One of the more modest was Helmholtz's, already mentioned in
Chapter V, with only twenty-four notes in the octave. 13 it fol-
lowed a suggestion by Euler in 1739 that each manual be in the

11GiovanniBattistaDoni, Compendiodel trattato de' generi, e de' modi (Rome,
1635), Chap. 13.

12Shohe' Tanaka (in Vierteljahrsschrift fur Musikwissenschaft, VI [1890] , 85)
was in error in showing these notes of Doni as only a comma higher than
the lower note of the pair forming the semitone.

13Helmholtz, Sensations of Tone, p. 316 f.

Ill

TUNING AND TEMPERAMENT

Pythagorean tuning, the one manual a comma higher than the
other. General Thompson followed Doni's lead by having three
manuals on his Enharmonic Organ, with forty different pitches
in the octave. Henry Poole's Euharmonic Organ had only two
black keys on the keyboard; but through a series of eleven ped-
als all the notes could be transposed into five sharp and five flat
keys, giving fifty distinct pitches in the octave.

Liston's organ also relied upon pedals to obtain a great vari-
ety of notes with the minimum number of keys. 14 With only
twelve keys to the octave, tuned in just intonation, he was able
by means of six pedals to add their enharmonic equivalents, thus
having twenty-four notes in his normal scale. These are shown
in Table 108. Then by three acute pedals all these notes could
be raised in pitch by a comma. Two grave pedals similarly
lowered nine or eleven of the normal notes by a comma. Thus
Liston had a total of fifty -nine pitches available.

Of Liston's fifty-nine notes, there were ten pairs, such as
DId0-C*~2, which differed by the schisma, 2 cents. Further-
more, Cx-3 and E#~3 differed by only six cents from D*3 +1 and
F""1"3 respectively, and could be considered equivalent pairs also.
Thus there were essentially only 47 separate pitches. These in-
cluded four larger intervals: between C and C*~2 and between
Cx"4 and D"1 there were two commas; between E#_1 and F#~'
and between A#~ and B there were three. If these larger in-
tervals had been divided, the octave would have contained 43 +
2x2 + 2x3 = 53 commas, which is the number one might have

Table 108. Liston's Enharmonic Organ

B#-3 Fx-3 Cx-3

G*-2 D*-2 A#-2 E*-2

A-1 E-1 B"1 F#_1 C#_1

Bb° Fo co QO Do

^b-t-1 qD+i Db+i ^b+i e^1

nbb +2 Tpb +2

14Henry Liston, An Essay upon Perfect Intonation (Edinburgh, 1812), pp. 3-7,
33-40.

112

MULTIPLE DIVISION

anticipated. These "commas" are not all the same size. The
ditonic comma does not occur at all except as the sum of the
syntonic comma and the schisma. The syntonic comma is, as is
evident from the scheme of pedals, the most common interval.
But intervals of 20 cents, as D*~2 -E*30 , and of 26 cents, as
G+1 -G*~3 , also occur.

More ambitious was Steiner's system. ° For the key of C
he used 12 notes in just intonation, symmetrically arranged in
three groups of 4 notes each. But these could be transposed me-
chanically into any of 12 different keys, the keynotes being tuned
by perfect fifths. Thus there were 144 notes, but only 45 distinct
pitches. Shohe Tanaka adopted Steiner's idea of having 12 key-
notes in Pythagorean tuning, for mechanical transposition. But
he extended his keyboard to 26 different notes, as shown in Ta-
ble 10S. Of the 312 notes to the octave of Tanaka's "Transponir-
Harmonium" or "Enharmonium," there were only 70 unduplicated
pitches, no more than on an organ described by Ellis which had
a total of 14 x 11 or 154 notes to the octave, with 70 separate
pitches.

Table 109. Tanaka's Enharmonium

F#-2 c#-2 G#-2 D*-2 A#-2 E#-2

G"1 D"1 A-1 E-1 B"1 F#_1 C*"1

Bb° Fo co Go Do Ao £o

Qb+i dd+1 Ab+1 Eb+1 Bb+1 F+1

Equal Divisions

With Tanaka's Enharmonium we may safely drop the subject
of just intonation extended. The theory is simple enough: pro-
vide at least four sets of notes, each set being in Pythagorean
tuning and forming just major thirds with the notes in another
set; construct a keyboard upon which these notes may be played
with the minimum of inconvenience. Only in the design of the
keyboards did the inventors show their ingenuity, an ingenuity
that might better have been devoted to something more practical.

15Tanaka, op. cit., pp. 18 f. and 23 ff.

113

TUNING AND TEMPERAMENT

The other direction in which multiple division developed had
far greater possibilities. This was the division of the octave into
more than twelve acoustically equal parts. " Any regular sys-
tem of tuning — a system constructed on a fixed value of the
fifth — will eventually reach a point where its "comma," the er-
ror for the enharmonic equivalent of the keynote, is small enough
to be disregarded. Thus we have closed systems that agree
more or less closely with the various types of meantone temper-
ament, etc.

If the Pythagorean tuning is extended to 17 notes, an interval
of 66 cents is formed — a doubly diminished third, as Ax-C. Di-
vided among 17 notes, the deficit is about 4 cents, the amount by
which each fifth must be raised to have a closed system. The
fifth (now taken as 10/17 octave) contains 706 cents, being raised
by about the same amount that it is lowered in the Silbermann
variety of meantone temperament. The major third (6/17 octave)
contains 423 cents, being more than twice as sharp as it is in
equal temperament, and the minor third is correspondingly very
flat. If we take 5 parts for the third, this becomes a neutral third
of 353 cents, such as the thirds found in some scales of the Orient.

In the 17 -division, the tone is composed of 3 equal parts, of
which the diatonic semitone comprises 1 part and the chromatic
semitone 2 parts. Since the diatonic semitone, 70 cents, is even
smaller than in the Pythagorean tuning, this system is well
adapted to melody. It is, of course, wholly unacceptable for
harmony because of its outsize thirds. It is notatedwith 5 sharps
and 5 flats only, D* and A* being considered the equivalent of
Fb and Cb, and Gb and Db the equivalent of E# and B#» The 17-
division is the well-known Arabian scale of third-tones. *-1

A much more popular system is the 19-division. It arises in
much the same way as the 17 -division, except that, as in just in -

l"For the sake of completeness two smaller divisions should be mentioned:
the Javanese equal pentatonic and the Siamese equal heptatonic. For a
strange reference to the latter see J. Murray Barbour, "Nierop's Hacke-
bort," Musical Quarterly, XX (1934), 312-319.

* 'Joseph Sauveur ("Systeme general des intervalles des sons," M^moires de
l'academie royale des sciences, 1701, pp. 445 f.) made an early reference
to this scale, and of course it is discussed in all modern accounts of Arabian
theory.

114

MULTIPLE DIVISION

tonation, the diatonic semitone is considered the larger, with 2
parts to 1 for the chromatic semitone. Since the octave contains
5 tones and 2 semitones, it will have 5x3 + 2x2= 19 parts.
The history of the 19-division goes back to the middle of the six-
teenth century, whenZarlino and Salinas discussed, among types
of meantone tuning, one in which the fifth was tempered by 1/3
comma. Like the other two types (1/4 and 2/7 comma) it was in-
tended for a cembalo with 19 notes to the octave. ° Salinas'
claim as inventor has not been disputed. He was rather apolo-
getic concerning it, because of its greater deviation from pure
intervals than the other two. He apparently did not realize that
this could not be distinguished from an equal division into 19
parts, and that thus, as a closed system, it possessed a great
advantage. It can be notated with 6 sharps and 6 flats, Cb being
the equivalent of B* and E* of Fb.

We have plenty of evidence from past centuries of cembali
with 19 notes in the octave, for which this division would have
been the ideal tuning. Zarlino19 described such a cembalo that
Master Domenico Pesarese had made for him. Elsasz is fre-
quently but erroneously called the inventor of the 19-note cem-
balo, because his instrument is described in Praetorius' Syn-
tagma.

After having been neglected during the nineteenth century for
the more elaborate systems such as have been described in the
previous section of this chapter, the 19-division was revived in
the second quarter of the twentieth century. It has had eloquent
contemporary advocates in Ariel, Kornerup, and Yasser. Of all
these enthusiasts, Yasser has gone to the greatest pains to show
the construction of the system and its possibilities. He differs
radically from its other adherents, who have proposed it partly
for the sake of differentiating enharmonic pairs of notes, but
chiefly because its triads are more consonant than those of equal
temperament. Yasser holds that the harmony of Scriabin and the

*°See Chapter III for further discussion of the various equivalents of the cy-
clic multiple systems.

^Institutioni armoniche, p. 140.

^Joseph Yasser, A Theory of Evolving Tonality (New York, 1932).

115

TUNING AND TEMPERAMENT

tone-rows of Schonberg show an intuitive striving toward the 19-
division, since a scale as used should contain unequal divisions,
being a selection from an equal division of more parts. Thus
the Siamese scale of 7 equal parts is suitable for pentatonic
melodies; the ordinary 12-note chromatic scale, for heptatonic
melodies; and the 19-division for melodies built upon the 12-
note scale. Yasser's attempt to give a historical foundation is
so defective that his case emerges considerably weaker than if
he had presented his system simply from the speculative point
of view.

There does not seem to be much chance of the 19-division
coming into use in our day. Its thirds and fifths have been dis-
cussed in Chapter III. To modern ears, accustomed to the sharp
major thirds of equal temperament, the thirds of 379 cents, 1/3
comma flat, would sound insipid in the extreme. There would
seem to be a better chance for the acceptance of a system that
does not differ so markedly in its intervals from our own.

The 22 -division belongs next in our study of equal divisions.
It was not discussed by Sauveur, Romieu, or Drobisch. In fact,
Bosanquet did not even mention it in his comprehensive book on
temperament, although Opelt had treated it carefully twenty -five
years before. * But the following year Bosanquet contributed
an article to the Royal Society, "On the Hindoo Division of the
Octave." In it he referred to S. M. Tagore's Hindu Music and
an article in Fetis' Histoire generate. There the Hindoo scale
was said to consist of 22 small intervals called "S'rutis." If
these are considered equal, a new system arises with "practi-
cally perfect" major thirds (actually, being 381.5 cents, they are
almost 5 cents flat) and very sharp fifths (709 cents, or 7 cents
sharp). Riemann later was to include the 22-division in his dis-
cussion of various systems, and it is frequently mentioned today.
Unfortunately, the Hindoo theory does not make the S'rutis all
equal, but that does not prevent the division from finding an hon-
ored place among these others.

The thirds of the 22-division are better than those of the 19-
division, and its fifths are no worse. However, it is not so good

21F. W. Opelt, Allgemeine Theorie der Musik, Chap. IV.
116

MULTIPLE DIVISION

a system for the performance of European music. The difficulty
lies in the formation of the major third. The fifth is taken as
13/22 octave, whence the tone has 4 parts and the ditone, 8. But
8/22 octave is 436 cents, an impossibly high value. Hence the
major third must be only 7 parts, or 381.5 cents. This means
that D# is taken as the major third above C, and Fb (or Cx) as
the third above B. This is an awkward feature, but one that we
shall run into with most of these equal divisions. It is not or-
dinarily possible to retain our ideas of tone relations while mak-
ing a division of the octave that will provide good fifths and thirds.

The 24 -division has the same good fifths and sharp thirds as
the 12-division, and the deviations for the 29-division are very
similar, but with plus and minus signs reversed. Both the 25-
and the 28-divisions have good thirds and quite poor fifths. So
none of these four divisions is of great import. The 24 -division
does have its place, as a possible realization of Aristoxenus'
theory that the enharmonic diesis is a true quarter tone, the half
of the equal semitone. Kircher22 presented it as such, together
with a geometrical method of obtaining the quarter tones on the
monochordo Rossi2** later gave the string- lengths for equal
quarter tones, and Neidhardt offered a similar table many years
afterwards. 24 The 29-division has its place as a member in the
series that contains the 17 -division, but that fact does not im-
prove the quality of its thirds.

The next system of importance is the 31-division. It is the
most ancient of them all and well worth the attention that has
been given to it. Observe that 31 logically follows 19 in the Fi-
bonacci series: 5, 7, 12, 19, 31, 50, 81, This system was

first described by Vicentino25 in 1555, as the method of tuning
his Archicembalo . In theory this was constructed in an attempt
to reconcile the ideas of the ancient Greeks with those of six-
teenth century practice. In reality it was a clever method for
extending the usual meantone temperament of 1/4 comma until

22Musurgia universalis, I, p. 208.

2^Sistema musico, p. 102.

24J. G. Neidhardt, Sectio canonis harmonici, p. 31.

25L'antica musica, Book 5, Chaps. 3-5.

117

TUNING AND TEMPERAMENT

it formed practically a closed system.

The Archicembalo contained six ranks of keys, of which the
first two represented the ordinary harpsichord keyboard with 7
natural keys, 3 sharps, and 2 flats. The third "order" contained
4 more sharps and 3 flats. The fourth order continued the flat
succession with 7 more keys, and the fifth added 5 more sharps.
(The sixth order is in tune with the first.) Thus all the notes
would lie in a succession of fifths from G*30 to Ax, and the cir-
cle would be completed by taking Ex as equivalent to G^b or C^b
to Ax. (Vicentino himself gave a second tuning to the fourth or-
der that showed that he considered the above to be equivalent
pitches.)

Vicentino specified that the first three orders of the Archi-
cembalo should be tuned "justly with the temperament of the
flattened fifth, according to the usage and tuning common to all
the keyboard instruments, as organs, cembali, clavichords, and
the like." But the other three orders may be tuned "with the
perfect fifth" to the first three orders. For example, the G of
the fourth order (that is, Abb) is to be a perfect fifth above the C
of the first order. It must be admitted that this part of Vicen-
tino' s scheme does not seem to make sense.

If we ignore this puzzling doctrine of the perfect fifth, we
have a logical system, formed by a complete sequence of 31 tem-
pered fifths. The amount of tempering is not specified, but was
to be the same as that of common practice. The common prac-
tice was the ordinary meantone temperament, in which major
thirds are perfect. This is undoubtedly what Vicentino used.

By logarithms Christian Huyghens^" showed that the 31-di-
vision does not differ perceptibly from the 1/4-comma tempera-
ment. More specifically he said: "The fifth of our division is
no more than 1/110 comma higher than the tempered fifths, which
difference is entirely imperceptible; but which would render that
consonance so much the more perfect." Riemann^' was con-
fused by this remark, not realizing that Huyghens meant that this
fifth was 1/110 comma higher than a fifth tempered by 1/4 comma.

26"Novus cyclus harmonious," Opera varia (Leyden, 1724), pp. 747-754.

2'Geschichte der Musiktheorie, p. 359.
118

MULTIPLE DIVISION

The difference between the logarithm of the meantone fifth,
.174725011, and that of 21B/3\ .1757916100, is .0000491089, which
is quite close to 1/110 of the logarithm of the syntonic comma,
.0053951317.

Tanaka^S and Riemann have described Gonzaga's harpsichord
intheMuseoCivico inBologna, dated 1606. Essentially the same
as Vicentino's instrument, its arrangement of notes is somewhat
different, the second row, for example, consisting solely of
sharped notes, instead of 3 sharps and 2 flats. Father Scipione
Stella's eight-manual harpsichord also resembled Vicentino's,
but had a couple of manuals duplicated to facilitate the execu-
tion.29

An improved version of Vicentino's Archicembalo was Colon -
na's 6 -manual Sambuca Lincea.30 The difficulty with Vicentino's
system was the unsystematic arrangement of the second and
third orders. Both C# and Eb, for example, were in the second
order, while Db and D* were in the third. If the instrument was
to be considered merely an extension of an ordinary cembalo
with twelve notes in the octave, such an arrangement was no
doubt good enough. But, for its complete possibilities to be
available, any such instrument needs what Bosanquet called a
"generalized keyboard."

Colonna came close to supplying this lack. Each of his or-
ders contained seven notes, and was 1/5 tone above the preced-
ing order. In our notation, the notes between C in the first or-
der and D in the sixth would be DDb, C*, Db, and Cx„ Colonna' s
notation for them was Cx, C#, DD, and C*, respectively. This is
very clumsy; but his idea of the division was entirely correct,
as can be seen from the scales he listed as examples of the ca-
pabilities of the instrument. He included such remote major
keys as Cb, A#, Ebb, and G# - all of course with his peculiar
notation.

28Shoh£ Tanaka, in Vierteljahrsschrift fur Musikwissenschaft, VI (1890),
pp. 74 f. ~

29Fabio Colonna, La sambuca lincea (Naples, 1618), p. 6.
3Qlbid., passim.

119

TUNING AND TEMPERAMENT

The germ of the 31-division lay in the contention of Marchet-
tus of Padua that a tone could be divided into five parts. After
Vicentino, Salinas and Mersenne discussed the system without
realizing its value. Hizler31 referred to a 31-note octave, but
used in practice only 13 notes, having both a D# and an E".
Rossi3^ anticipated Huyghens in obtaining by logarithms the
string- lengths for the 31-division, but did not call attention to
the fact that its pitches were so close to those of the meantone
temperament which he also presented. (With A at 41472, his
meantone E was 27734, the 31-division E, 27730.) Gallimard33
was to follow Huyghen's lead in comparing the logarithms of the
two temperaments. Van Blankenburg3"* was to use the 31-divi-
sion as a sort of tuning measure, much as Sauveur used the 43-
division and Mercator the 53-division. According to van Blank -
enburg, Neidhardt's equal temperament was full of "young wolves,
each 1/3 of the large wolf," because the major third of equal
temperament contains 10 1/3 parts instead of the 10 parts of the
31-division.

The string-lengths for the 31-division were also given by
Ambrose Warren,3^ for the octave 8000.0 to 4000.0. Warren
showed how this temperament could be applied to the fingerboard
of the violin, for a string 13 inches long.

For obtaining the 31-division mechanically, Rossi recom-
mended the mesolabium. Salinas, Zarlino, and Philander have
stated that the mesolabium could be used for finding an unlimited
number of geometrical means between two lines, provided the
number of parallelograms was increased correspondingly. Per-
haps so, but Rossi3** was undoubtedly correct in saying that "in
dividing the octave into 31 parts you will experience greater dif-

31Daniel Hizler, Extract aus der neuen Musica Oder Singkunst (Niirnberg,
1623), p. 31.

^^Sistema musico, pp. 86, 64.

33J. E. Gallimard, L'arithmetique des musiciens (Paris, 1754), Table XVI,
p. 25.

3**Elementa musica, p. 115.

35The Tonometer (London, 1725), table at end of book.

3"Sistema musico, p. 111.

120

MULTIPLE DIVISION

ficulty because of the great number of rectangles," and Mer-
senne37 said flatly that it "is of no use except for finding two
means between two given lines."

Romieu^° included the 31 -division among those for which he
had obtained correspondences, calling it a temperament of 2/9
comma. This is not very close, for 1/4 - 1/110 = 53/220. (Dro-
bisch's 74-division is the real 2/9-comma temperament.) It is
possible that writers before Romieu had this tuning in mind when
they wrote about the 2/9-comma temperament. Printzy^ for ex-
ample, spoke of a "still earlier" temperament that took 2/9
comma from each fifth. Earlier, perhaps, than Zarlino's 2/7
comma, which he had been discussing previously. But Lemme
Rossi, who gave a detailed treatment to the 2/9-comma tuning,
did not identify it with the 31 -division.

The 34-division is a positive system, like the 22-division.
That is, its fifth of 706 cents is larger than the perfect fifth, be-
ing the same size as for the 17 -division. Its third is about 2
cents sharp. Thus it provides slightly greater consonance than
the 31 -division. But, like the 22-division, it has remained one
of the stepchildren of multiple division, largely because it is in
a series for which ordinary notation cannot be used. There is a
surprising mention of the 34-division by Cyriac Schneegass in
1591 (see Chapter III), but his own monochord came closer to
the 2/9-comma division. Bosanquet had indicated the relation
between the 22- and 34-divisions, and had praised the 56- and
87-divisions also as similar systems. Opelt, too, has included
it in his fairly short list.

The 36 -division has little to recommend it, although its string-
lengths were worked out by Berlin,'*" and Appun and Oettingen
both found it worth describing.'*-'-

The 41-division has excellent fifths (702.4 cents), but thirds

3'Harmonie universelle, p. 224.

JOIn Memoires de 1 academie royale des sciences, 1758, p. 837.

^"phrynis Mytilenaeus oder der satyrische Componist, p. 88.

40Johann Daniel Berlin, Anleitung zur Tonometrie (Copenhagen and Leipzig,
1767), pp. 26-27.

41Hugo Riemann, Populare Darstellung der Akustik (Berlin, 1896), p. 138.

121

TUNING AND TEMPERAMENT

(380.5) that are almost six cents flat, being in this latter respect
inferior to the 31- and 34 -divisions. It occurs in a worthy se-
ries: 12, 17, 29, 41, 53, ... . This system was not singled out
by any of the earlier writers, but received considerable atten-
tion from such nineteenth century theorists as Delezenne, Dro-
bisch, andBosanquet. Paul von Janko"*^ set himself the task of as-
certaining the best system between 12 and 53 divisions, and chose
the 41 -division. Rather naively, he concluded he had discovered
this system, since Riemann had not mentioned it!

The 43-division is associated with the name of Sauveur, 4^
who used its intervals (Merides) as a unit of musical measure.
The Merides were divided into seven parts called Eptamerides.
For more subtle distinctions, Sauveur suggested using Decam-
erides, 10 of which comprised one Eptameride. But he did not
use the Decamerides in practice. Thus there were 43 x 7 = 301
Eptamerides in the octave, or 3010 Decamerides. Since .30103
is the common logarithm of 2, it is possible to convert directly
from logarithms to Eptamerides by dropping the decimal point
and all but the first three digits of the logarithm.

The 43-division is a closed system approximating the 1/5-
comma variety of meantone temperament, which, as we saw in
Chapter III, had been mentioned by Verheijen and Rossi. Its
thirds and fifths have an equal and opposite error of slightly over
four cents, thus making it somewhat inferior to the 34 -division,
although the equality of the error may have some weight in rank-
ing the two systems. Since 43 is a number occurring in a useful
series for multiple division — 12, 19, 31, 43, 55, ... — this divi-
sion was treated by Romieu, Opelt, Drobisch, and Bosanquet.

The 50-division need not detain us long. It may be thought of
as an octave composed of ditonic commas, since 1200 r 24 = 50.
It was advocated by Henfling in 1710 and criticized by Sauveur44
the following year. A century later Opelt was to mention it.

42"Uber mehr als zwolfstufige gleichschwebende Temperaturen," Beitrage
zur Akustik und Musikwissenschaft, 1901, pp. 6-12.

43joseph Sauveur, in Memoires de l'academie royale des sciences, 1701, pp.
403-498.

44joseph Sauveur, "Table generate des systemes temp£re's de musique,"
Mlmoires de l'academie royale des sciences, 1711, p. 406 f.

122

MULTIPLE DIVISION

Bosanquet has included it as a member of the series: 12, 19,
31, 50, .... This division shows no improvement over the 31-
division. Its fifths have about the same value as those of the
latter, and its thirds are flatter than the latter' s were sharp.
Kornerup^ has waxed lyrical in its praise, as a closed system
corresponding to Zarlino's 2/7 -comma meantone temperament.
He showed that the value for Zarlino's chromatic semitone
(70.6724 cents) came very close to the mean of the chromatic
semitones for the 19- and 31 -divisions (70.2886), and might have
added that this similarity extends throughout, since all three are
regular systems. He found that the greatest deviation of the
2/7 -comma tuning from the 50-division is a little over three cents,
and is much less for most notes. We shall have more to say
later about the special part of Kornerup's theory that has caused
him to overvalue this system.

The most important system after the 31- is the 53-division.
In theory it is also the most ancient. According to Boethius, "
Pythagoras' disciple Philolaus held that, since the tone is divis-
ible into minor semitones and a comma, and since the semitone
is divisible into two diaschismata, the tone is then divisible into
four diaschismata plus a comma. If, now, the diaschisma is taken
as two commas exactly, the tone is divided into nine commas.
(Note what was said about the ditonic comma in connection with
the 50-division.)

This dictum about the number of commas in a tone was one
of the most persistent parts of the Pythagorean system. Writers
in the early sixteenth century sometimes mentioned the fact that
there are nine commas in a tone, without giving any other tuning
lore. They probably included, however, the statement that the
diatonic semitone contains four commas, the chromatic semitone,
five. Amusingly enough, after just intonation became the ideal,
writers continued to talk about commas; but now it was the chro-
matic semitone that contained four commas, the diatonic semi-
tone, five.

Since the Pythagorean diatonic semitone contains 90 cents,

45Thorvald Kornerup, Das Tonsystem des Italieners Zarlino (Copenhagen,
1930).

46A. M. S. Boethius, De institutione musica, Book 3, Chap. 8.

123

TUNING AND TEMPERAMENT

and the chromatic, 114, their ratio is 3 3/4:4 3/4, or approxi-
mately 4:5. Similarly, if we choose the larger just chromatic
semitone of 92 cents and the smaller just diatonic semitone of
112 cents, the ratio will be 4 1/2:5 1/2, or, again, 4:5. But the
ratio might be taken as 5:6, giving rise to the 67-division dis-
cussed below. The comma, taken as 1/9 Pythagorean tone, would
have a mean value of 22.7 cents, lying between the syntonic and
the ditonic commas.

If there are 9 commas in a tone, the octave contains 5x9 +
2 x 4 = 53 commas — provided we are thinking in terms of the
Pythagorean tuning. If we are thinking in terms of just intona-
tion, with a large diatonic semitone, there will be 5x9 + 2x5 =
55 commas. Thus the 55-division has received attention also.

There are several advantages to the 53-division. Its fifths
are practically perfect (.1 cent flat), so that it is unnecessary to
use a monochord for tuning. Its thirds are very slightly flat
(1.4 cents). However, since it is a positive system, with fifths
sharper than those of equal temperament, the pure major third
above C is F", with 17 parts, whereas C-E represents the Pyth-
agorean third, with 18 parts. This would be confusing to the
performer.

After the time of the Greeks, the history of the 53-division
takes us to China, where the Pythagorean tuning had been known
for many centuries, probably since the invasion of Alexander the
Great. In 1713 it was confirmed as the official scale, however
widely instrumental tunings may have differed from it in practice.

One of the most remarkable of the early Chinese theorists
was King FSng, who, according to Courant, "calculated ex-
actly the proportional numbers to 60 111," that is, he extended the
Pythagorean system to 60 notes. These results were published
by Seu-ma Pyeou, who died in 306 A. D. King FSng observed
that the 54th note was almost identical with the first note. Cou-
rant's figures are 177,147 for the first; 176,777 for the 54th.

Seventeenth century European theorists who referred defin-
itely to this system include Mersenne and Kircher. Tanaka

^Maurice Courant, "Chine et Core"e," Encyclopedic de la musique et dic-
tionnaire du conservatoire (Paris, 1913), Part I, Vol. I, p. 88.

124

MULTIPLE DIVISION

mentioned Kircher's name in this connection, thus differing from
the majority of his contemporaries, who ascribed the system to
Mercator. According to Holder,48 Nicholas Mercator had "de-
duced an ingenious Invention of finding and applying a least Com-
mon Measure to all Harmonic Intervals, not precisely perfect,
but very near it." This was the division into 53 commas. There
is no evidence, in Holder's account, that Mercator intended this
system to be used on an instrument. It was to be merely a
"Common Measure."

Of 25 systems that Sauveur discussed, only two, the 17- and
53-divisions, were positive. He was unable to appreciate the
splendid value of the thirds of the latter, since, according to his
theory, its thirds would have to be as large as Pythagorean thirds.
Romieu did not even mention this system. Drobisch, too, did not
at first (1853) appreciate the 53-division, discarding it because
of its sharp thirds. But two years later he re-evaluated both the
41- and the 53-divisions, showing that a just major scale could
be obtained with them by using C D Fb G Bbb Cb C.49

The stage was thus set for Bosanquet' s detailed study of mul-
tiple division, which culminated in his invention of the "gener-
alized keyboard" for regular systems. In his article in the Royal
Society's Proceedings, 1874-75, Bosanquet gave a clear and com-
prehensive treatment of regular systems, both positive and neg-
ative, with a possible notation for them. He showed how various
systems could be applied to his keyboard, especially the 53- and
118 -divisions. In his symmetrical arrangement, 84 keys were
needed for the 53 different notes in the octave. Obviously, then,
Bosanquet' s name should be singled out for especial mention,
since he applied the system to an enharmonic harmonium and did
not simply discuss it as his predecessors had done.

As has been noted above, the 55-division is the negative coun-
terpart of the 53-division, thus having the advantage that ordi-
nary notation can be used. That is its only advantage, for its
fifths (698.2 cents) are no better than those of the 43-division,

48william Holder, Treatise . . .of Harmony, p. 79.

49m, W. Drobisch, "Uber musikalische Tonbestimmung und Temperatur,"
Abhandlungen der mathematisch-physischen Classe der koniglich s'achsis-
chen Gesellschaft der Wissenschaften, IV (1855), 82-86.

125

TUNING AND TEMPERAMENT

and its thirds (392.7 cents) are inferior to the latter' s. Sauveur
devoted considerable space to this system, saying it was "fol-
lowed by the musicians." This is a reasonable statement, for
this system corresponds closely to the 1/6 -comma variety of
meantone temperament favored by Silbermann. Thus we have
confirmation from France of the spread of this method.

Romieu showed the correspondence between the 55-division
and the 1/6-comma tuning, and adopted the latter for his "tem-
perament anacratique. " He referred to Sauveur, and also to
Ramarin's system as given in Kircher. Mattheson^l presented
this division from Johann Beer's Schola phonologica, saying that
it required "that an octave should have 55 commas, but no ma-
jor or minor tones."

Sorge, after disapproving of the ordinary 1/4 -comma mean-
tone, continued: "I am better pleased by the famous Capell-
meister Telemann's system of intervals, in which the octave is
divided into 55 geometrical parts (commas), that grow smaller
from step to step." 52 Sorge explained that in its complete state
it could not be used on the clavier; but it might be applied to the
violin and to certain wind instruments, and was easiest for singers.

^Correspondences between multiple divisions and temperaments by fractional
parts of the syntonic comma can be worked out by continued fractions.
When the temperament of the fifth is 1/2 comma, the octave contains 26
parts. If d is the denominator of the fractional part of the comma (21.5
cents), the following formula gives the parts in the octave for 2<d<:ll:
Sd = 7 + 12 |~114d - 150.51 ' wnere the expression in brackets is to betaken

[l 2(2 1.5 - 2d)J
to the nearest integer. The list of correspondences is: 1/3 comma, 19
parts; 1/4 comma, 31 parts; 1/5 comma, 43 parts; 1/6 comma, 55 parts;
1/7 comma, 91 parts; 1/8 comma, 139 parts; 1/9 comma, 247 parts; 1/10
comma, 499 parts. Temperaments in which the numerator of the fraction
is 2 are formed as follows: 1+1 _ 2 comma, 26 + 19 = 45 parts; 1 + 1 _ 2

2T~3 " 5 3 + 4 " 7

comma, 19 + 31 = 50 parts; 1+1 _ 2 comma, 31 + 43 = 74 parts; 1 + 1 _ 2

4 + 5 ~ 9 5 + 6 "11

comma, 43 + 55 = 98 parts, etc.

51J. Mattheson, Critica musica (Hamburg, 1722-25), II, 73 f.

"^Georg Andreas Sorge, Gesprach zwischen einem Musico theoretico und
einem Studioso musices, pp. 51 f.

126

MULTIPLE DIVISION

William Jackson*^ found that the octave consists of 55 10/12
syntonic commas, or 670 units of 1/12 comma. He might well
have assumed the octave to contain 56 commas precisely, since
this is a fairly good division. A half century after Jackson, an
anonymous work printed in Holland*^ stated that the ratio 81:80
is contained 56 times in the octave, but did not advocate this as
a system of multiple division. Bosanquet mentioned the 56 -divi-
sion. It has excellent thirds, being 1 cent flat, as in the 28-di-
vision. Its fifths are 5 cents sharp.

The 58 -division is also positive, its fifths being 2 cents sharp,
as in the 29 -division, and its thirds being 7 cents sharp. This
is the division that is at the base of Dom Bedos' temperament, °
although he chose the pitches for his monochord somewhat ir-
regularly from it.

There are only a few other systems that should be mentioned.
The 65-division has splendid fifths (.5 cent flat) and slightly
sharp thirds (1.4 cents sharp). The 84-division, on the other
hand, has only average fifths (2 cents flat), but excellent thirds
(.6 cent flat). The 87-division has slightly sharp fifths (1.4 cents
sharp), and practically perfect thirds (.1 cent flat). The 118-di-
vision has both fifths and thirds that are superlative (.5 cent flat
and o2 sharp respectively).

The above four systems excel all others with more than 53
parts in the octave. But the specialists in multiple division have
not always appreciated them. Sauveur, for example, discussed
the 67-, 74-, 98-, 105-, 112-,and 117 -divisions, as well as others
that are no better than they, but did not mention any of the four
systems in the previous paragraph. Romieu did not discuss any
systems beyond the 55 -division, but would have approved the 67-,
79-, and 91-divisions. Drobisch particularly favored the 74-di-
vision among systems that formed the major third regularly, as

53A Scheme Demonstrating the Perfection and Harmony of Sounds (London,
1726 [?]), chart.

^Exposition de quelques nouvelles vues mathematiques dans la theorie de la
musique (Amsterdam, 1760), p. 28.

55Dom Francois Bedos de Celles, L'art du facteur d'orgues, 2nd Part (Paris,
1770); facsimile ed. (Kassel, 1935), p. 430.

127

TUNING AND TEMPERAMENT

C-E; among those that used C-F13 as a major third, he mentioned
the 65-, 70-, 77-, 89-, and 94-divisions, and found the 53- and
118-divisions best of all. Bosanquet, praising most highly the
53- and 118-divisions, had kind words for the 56-, 65-, and 87-
divisions also.

Theory of Multiple Division

The reason for the divergent results obtained by these theo-
rists is that each had a different theory regarding acceptable di-
visions of the octave. Sauveur, although he did list two positive
systems, had no real understanding of divisions in which C-F*3
could be a major third„ To him, the diatonic semitone was the
larger: the problem of temperament was to decide upon a def-
inite ratio between the diatonic and chromatic semitones, and
that would automatically give a particular division of the octave.
If, for example, the ratio is 3:2, there are 5x7 + 2x4 = 43
parts; if 5:4, there are 5x9 + 2x5 = 55 parts. We have pointed
out above that only the first of these divisions is at all satisfac-
tory. Let us see what the limit of the value of the fifth would be
if the (n + l):n series were extended indefinitely. The fifth is
(7n + 4)/(12n + 7) octave, and its limit, as n 5» ©<=> , is 7/12 oc-
tave; that is, the fifth of equal temperament. The third, similarly,
approaches 1/3 octave. Therefore, the farther the series goes,
the better become its fifths, the poorer its thirds. This would
seem, then, to be an inferior theory.

In other divisions listed by Sauveur the difference between
the two sizes of semitone was two, three, or even four parts.
Here, again, the fifth eventually comes close to 7/12 octave and
the third to 3/12 octave. Romieu followed Sauveur' s theory. To
an extent so did Bosanquet. But the latter added the theory of
positive systems. The primary positive system is 17, 29, 41,

53, 65, 77, 89, Here the fifth can be expressed as (7n + 3)/

(12n + 5) octave. Just as in the negative systems above, the limit
of this ratio is 7/12 octave,, For the 53- and 65-divisions the
fifths are practically perfect; the thirds of these divisions have
approximately equal, but opposite, deviations. This suggests a

128

MULTIPLE DIVISION

secondary positive system, the mean between the former two:
118, with both fifths and thirds well-nigh perfect. But there is
nothing in these series themselves to facilitate choosing the best
division or the two best. That had to be ascertained by compar-
ing the intervals in the various divisions after they had been
chosen. Again it would seem as if there were an arbitrary fac-
tor present.

We have already spoken of Kornerup and his fondness for the
50-division.56 His "golden" system of music was suggested by
a study made by P.S. Wedell and N. P. J. Bertelsen in 1915. By
the method of least squares they obtained the following octave
series in which both the major third (5:4) and the augmented
sixth (that is, the minor seventh, 7:4) approach their pure val-
ues: 3, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, These of

course are "golden" numbers, the law of the series being Sn =
sn-i + sn-2- As n ><*>, s >a[5-1 =.61803398, a

s~n 2

an+ i
ratio which Kornerup called go . It is this ratio which is used in
the golden section of a line, where (1 - co )/co = 6U, and which
Kornerup used as the basis of his tuning system. By rather
simple arithmetic we find that the golden fifth is (15 - /\[5)/22 oc-
tave, or 696.2144738 cents. The golden third is 384.8579 cents,
only a fair approximation, since the pure third is 386.3137 cents.
Therefore, even if the series is continued indefinitely, the fifth
will never be less than about 6 cents, nor the third than 1.5 cents
flat. Since we have already observed several systems with bet-
ter thirds and fifths than this, it would seem as if the golden
system is an ignis fatuus.

Drobisch57 gave an interesting formula which combined Bo-
sanquet's primary and secondary positive systems. The fifth of
these systems will be: (7n - l)/2(6n - 1). For odd values of n,
the octave contains 6n - 1 parts; for even values, twice as many.

"'"See Thorvald Kornerup, Das goldene Tonsystem als Fundament des theo-
retischen Akustik (Copenhagen, 1935).

"* 'In Abhandlungen der mathematisch-physischen Classe der koniglich sach-
sischen Gesellschaft der Wissenschaften, IV (1855), 79 f.

129

TUNING AND TEMPERAMENT

Hence he obtained the series (with n ranging from 4 through 15) :
46, 29, 70, 41, 94, 53, 118, 65, 142, 77, 166, 89.

Somewhat more general was Drobisch's attempt to find a di-
vision of the octave that would insure a good value for the fifth.
He expressed the ratio of the fifth (log 3/2) to the octave (log 2)
as a decimal, .5849625, or as a fraction, 46797/80000. From
this ratio, by binary continued fractions, he obtained the series

2, 5, 12, 41, 53, 306, 665, [15601] , Next he found all the

powers of 3/2 from the 13th to the 53rd, in order to ascertain
which approach a pure octave. This should have checked closely
with his previous list, to which 17 and 29 would be semi-conver-
gents. This, however, is his complete list: 17, 19, 22, 29, 31,
41, 43, 46, 51, 53. Having eliminated all positive divisions (those
with raised fifths), he still had 19, 31, and 43 to add to his prev-
ious list.

Although the 50-division did not appear on either list, Dro-
bisch anticipated Kornerup by showing that its fifth lies almost
exactly between the fifths of the 19- and 31 -divisions. After
these promising beginnings, he went off at a tangent by trying to
find, by least squares, the value of the fifth that would produce
the best values for five different intervals. Then, again using
continued fractions, he found that successive approximations to

this value (.5810541) form the series: 2, 5, 7, 12, 31, 74,

This is why the 74-division had an especial appeal for him.

Drobisch's continued fractions were the first really scien-
tific method of dividing the octave with regard to the principal
consonances, the thirds and the fifths. The difficulty with it is
that there are three magnitudes to be compared (third, fifth, and
octave), but only one ratio (third to octave, fifth to octave, pos-
sibly third to fifth) can be approximated by binary continued frac-
tions. If we must choose a single ratio, it is better to use that
of the fifth to the octave, as Drobisch did, since the third may be
expressed in terms of the fifth. But the usual formula, T = 4F -
20, is valid only through O = 12. We have already noted that as
fine a musical theorist as Sauveur failed to appreciate the 53-
division, since he used the above formula and obtained a third
that was one part large. Since the syntonic comma is about 1/56

130

MULTIPLE DIVISION

octave, this formula will fail to give a correct number of parts
for the third for any octave division greater than 28. Thus if
O = 41, and F = 24, the formula makes T = 4 x 24 - 2 x 41 = 14,
whereas the correct value is 13. If O = 665, and F = 389, T =
4 x 389 - 2 x 665 = 226, instead of 214. Knowing the value of the
comma, we can correct our formula to read: T = 4F - 20 - TO"].

[_56j
But even this would only by accident give a value for the third
with as small a deviation as that for the fifth in the same divi-
sion., What is needed is a method that will approach the just val-
ues for third and fifth simultaneously.

The desired solution can be obtained only by ternary con-
tinued fractions, which are a means by which the ratios of three
numbers may be approximated simultaneously, just as the ra-
tios of two numbers may be approximated by binary continued
fractions. When the ordinary or Jacobi ternary continued frac-
tions are applied to the logarithms of the major third (5:4), per-
fect fifth (3:2), and octave (2:1), the octave divisions will be:
3, 25, 28, 31, 87, 817,

There are two serious faults in these results. In the first
place, the expansion converges too rapidly, and we are interested
chiefly in small values, those for which the octave has fewer than
100 parts. In the second place, the first few terms are foreign
to every other proposed solution, such as those by Sauveur and
Drobisch on previous pages.

To insure slow convergence, a mixed expansion was evolved,
which yields the following excellent series of octave divisions:

3, 5, 7, 12, 19, 31, 34, 53, 87, 118, 559, 612, 58 The only

serious omission is the Hindoo division, with 22 parts in the oc-
tave o The last term shown above (612) was said by Bosanquet to
have been considered very good by Captain J. Herschel.

There is no record that Captain Herschel ever constructed
an experimental instrument with 612 separate pitches in the oc-
tave. Even if he had done so, it would have been a mechanical
monster, incapable of producing genuine music at the hands of a

^8J. Murray Barbour, "Music and Ternary Continued Fractions," American
Mathematical Monthly, LV (1948), 545-555.

131

TUNING AND TEMPERAMENT

performer. With the possible exception of the 19- and 22-divi-
sions, the same can be said of all these attempts at multiple di-
vision. Bosanquet's 53-division apparently was a success on the
harmonium he constructed with the "generalized keyboard." But
it, too, was cumbersome to play, and would have been very ex-
pensive if applied to a pipe organ or piano. Thus the mathemat-
ical theory, worked out laboriously by ternary continued frac-
tions, remains theory and nothing more. The practice for the
past five hundred years has favored almost exclusively systems
with only twelve different pitches in the octave. There seems no
immediate prospect of that practice being discarded in favor of
any system of multiple division.

132

Chapter VII. IRREGULAR SYSTEMS1

If we accept Bosanquet's definition that a "regular" tuning sys-
tem is one in which every fifth, or every fifth save one, has the
same value, this would include the Pythagorean tuning, equal tem-
perament, and the several varieties of the meantone tempera-
ment, as well as equal divisions with more than twelve notes in
the octave. With the addition of just intonation, it would seem as
if this covered the ground pretty thoroughly. There are, however,
a great many tuning systems that do not fall into any of the above-
mentioned classes. At first glance these irregular systems pre-
sent a bewildering variety. But some of them have been offered
by their sponsors as modifications of existing tuning systems,
and others, although not so designated, are also closely related
to regular systems. In fact, it is possible, by making the bounds
sufficiently elastic, to fit every one of these irregular systems
into one or another of certain subclasses. So that, unless we re-
tain Bosanquet's strict definition, there is no such thing as an
irregular system — one that is wholly a law unto itself!

Our first group of irregular temperaments consists of modi-
fications of the meantone temperament. The meantone wolf fifth
is 35 cents sharp. The simplest modification of this tempera-
ment is to divide this excess equally between the fifths C*-G^
and G*(Ab)-Eb (see Table 110). This is the modification gener-

Table 110. Meantone Temperament with Two Sharp Fifths

Names C C# D Eb E F F# G G# A Bb B C

Cents 0 76 193 310 386 503 579 697 793 890 1007 1083 1200

MD. 17.2; S.D. 18.5

ally, but erroneously, ascribed to Schlick, and, according to Ellis,
still in use in England in the early nineteenth century. The G*
is now almost a comma sharper than in the pure 1/4-comma

*Fora condensed version of the material in this chapter, see J. Murray Bar-
bour, "Irregular Systems of Temperament," Journal of the American Mu-
sicological Society, I (1948), 20-26.

TUNING AND TEMPERAMENT

temperament. The mean deviation is noticeably lower, but the
standard deviation is affected less0

Mersenne has included a discussion of the meantone temper-
ament with all his other tuning information. His account differs
slightly in the different works where it occurs. In the Harmonie
universelle (pp. 364 f.) he had made the fifths Eb-Bb-F perfect.
In the Cogitata physico-mathematica (p. 338) he asked the reader
to correct the "obvious errors" in the previous description. Here
he indicated simply that the wolf fifth will be G#-Eb. Perhaps
his real intent is to be found in Harmonicorum libri XH (p. 60),
where these two fifths are to be sharp, but not so sharp as the
wOlf fifth, which is still unusable. Mersenne said that the mean-
tone fifth is tempered "1/136, which is about 1/4 comma." This
is a gross misstatement, for the ratio given is larger than 1/2
comma. He probably meant 1/316, which is a reasonably close
value.

Mersenne' s improvements upon the regular meantone tem-
perament are worth showing, even if the second will be only an
approximation to what he had in mind. In the first temperament
(Table 111) the fifths ED-Bb and Bb-F are pure. For the second
(Table 112), note that the excess of the minor third G#(Ab)-F
over the third of equal temperament is 30 cents. Let us divide
this excess so that G#-ED bears only half of it, the other two
fifths one -quarter each.

Table 111. Mersenne's Improved Meantone Temperament, No. 1

Names C C# D Eb E F F# G G# A Bb B C

Cents 0 76 193 299 386 503 579 697 773 890 1001 1083 1200

M.D. 17.2; S.D. 17.7

Table 112. Mersenne's Improved Meantone Temperament, No. 2

Names C C* D Eb E F F# G G# A Bb B C

Cents 0 76 193 288 386 503 579 697 773 890 996 1083 1200

M.D. 15.3; S.D. 16.9

134

IRREGULAR SYSTEMS

In Mersenne's first improved meantone system, the mean de-
viation is no lower than for the temperament previously shown;
but the standard deviation is lower because more notes are in-
volved in the change. Mersenne's second improvement was the
pattern for a modification recommended by Rameau. Now Ra-
meau is noted chiefly in tuning history for his advocacy of equal
temperament. But he vacillated sufficiently in his adherence to
it to follow Huyghens in acclaiming as "the most perfect of all"
temperaments that in which "the fifth is diminished by the 1/4
part of a comma. ** But he was aware of the pitfalls of the mean-
tone temperament; for he showed that, if the tuning is begun on
C, G# will be a "minor comma," 2025/2048 too flat. The remain-
ing fifths, therefore, should be tuned "more just," "to regain the
minor comma that has been lost." It would be even better to be-
gin with C*, in order to spread the discrepancy over more notes.

This account sounds as if the excess should be divided equally
among the last five fifths. But, in a later paragraph, Rameau
declared that "the excess of the last two fifths and of the last
four or five major thirds is tolerable, not only because it is al-
most insensible, but also because it is found in modulations little
used/ Apparently the first three of the five fifths are not to be
so sharp as the final two fifths. Still later he recommended that
"the division begin on B*3, and only those fifths that follow B-F*
should be a little more just."

These directions are as vague as Mersenne's. In Table 113
the division is begun on bP as Rameau suggested. The fifths
from B to G* have been made pure, and the excess has been di-
vided equally between G*-D* and E^-B^.

Before considering a final, complicated modification of the
1/4-comma temperament, let us look at William Hawkes' im-

Table 113. Rameau's Modified Meantone Temperament

Names C C# D D# E F F# G G# A Bb B C

Cents 0 87 193 298 386 503 585 697 789 890 1007 1083 1200

M.D. 12.5; S.D. 14.0

2J. P. Rameau, Nouveau systeme de musique th^orique (Paris, 1726), pp. 107 ff.

135

TUNING AND TEMPERAMENT

provement upon the 1/5-comma temperament. This resembles
Mersenne's first modification. In it, according to John Farey,^
"each ascending fifth is flattened by one-fifth of a comma as the
instrument is tuned, except that the fifth above E*3 and the fifth
below G^ are directed to be tuned perfect." Farey continued:
u. . .but why these anomalies in the system are introduced I am
at a loss to guess, especially as G* is thereby made 1/5 comma
the worse by it," Hawkes' reason is perfectly valid — to dimin-
ish the wolf fifth by 2/5 comma, although it will still be 16 cents
sharp. The alteration results in a somewhat smaller deviation
than for the pure 1/5-comma temperament.

The most involved of all these temperaments was that of J. E.
Gallimard,4 who brought a knowledge of logarithms to bear upon
the problem, in order to obtain a subtly modified meantone tem-
perament. He expressed intervals for all the principal tuning
systems in Sauveur's Decamerides — four-place logarithms
without the decimal point. The first of his original tempera-
ments used the values of the 1/4-comma temperament for the
eight notes from B^ to B, If Gallimard had continued in this
fashion until the entire octave had been tuned, the final fifth
(D -B") would have borne the usual wolf, amounting to 103 Deca.
He split up this error by adding an ever-increasing amount to
each logarithm for the five fifths from B to A*. Thus there would
be a total of 1 + 2 + 3 + 4 + 5=15 parts to be divided into 103
Deca., or about 7 Deca. for each part. In cents, this means that
the first seven fifths have a value of 696 or 697 cents each, the
others 699, 702, 705, 708, 710 cents respectively. Gallimard has
pure thirds in all the principal triads of the keys of Fand C, and
the poorest thirds in the key of G^0 The third on G^ itself has
425 cents, practically a diesis sharp!

In Gallimard' s second temperament, the first eight notes were
tuned as in the previous temperament. But he distributed the
error among the other five fifths, proportional to the series 1,
3, 6, 10, 15; that is, to the series n(n-l)/2. The cents values for

3"On Music," Philosophical Magazine, XXVI (1806), 171-176.
^L'arithm^tique des musiciens, p. 26.

136

IRREGULAR SYSTEMS

these altered fifths are 698, 700, 704, 708, and 714» Here the
worst fifths are worse than in his first temperament, and this
error is reflected in a slightly higher deviation. His worst third,
G^-B^, is still a diesis sharp.

The deviations are still large for Gallimard's modification.
Had he been willing to use a modification of the 1/6-comma tem-
perament, with slightly sharp diatonic thirds, his system would
have been better. Modifications of the latter temperament are
to be found later in this chapter, by Young and Mercadier.

Arnold Schlick' s temperament^ deserves special honor, for
apparently he was the first writer in any country to describe a
temperament for each note of the chromatic octave. Shohe Ta-
naka and Hugo Riemann have broadcast the erroneous idea that
Schlick founded the meantone system. The former spoke of the
"exact instructions" that Schlick had given, and added, "In exact
language this will mean that each fifth is to be flattened by 1/4
comma."" This reads well, but is utter nonsense with relation
to what Schlick actually said. In place of "exact instructions" he
gave very indefinite rules that create a problem for us.

Beginning with F on the organ manual, the fifth F-C is to be
somewhat flat. This same rule is to be followed in tuning the
other "claves naturales" by fifths, making the octaves perfect.
As to the major thirds, Schlick said that "although they will all
be too high, it is necessary to make the three thirds C-E, F-A,
and G-B better, ... as much as the said thirds are better, so much
will G# be worse to E and B."

The tuning of the black keys is to be made similarly, tuning
upward by flat fifths from B to obtain F* and C#, and tuning down-
ward from F to obtain Bb and ED. The semitone between G and
A received special attention. As G* it was needed as the third
above E; as A13 it was also needed as the third below C. So
Schlick suggested a mean value for this note, directing that the
fifth AD-ED is to be somewhat larger than a perfect fifth.

Whatever Schlick' s system, it could not have been the mean-

kSpiegel der Orgelmacher und Organisten, in Monatshefte fur Musikge-
schichte, 1869, pp. 41 f.

"Shohe Tanaka, in Vierteljahrsschrift fur Musikwissenschaft, VI (1890), 62,

137

TUNING AND TEMPERAMENT

tone system as described so carefully by Tanaka; for it lacks
pure thirds. Schlick said definitely that "all will be too high."
Not even the diatonic thirds are to be pure, only made "better
than the rest."

What, then, was Schlick' s tuning method? All that can be said
with assurance is that it was an irregular system, lying some-
where between meantone and equal temperament. We cannot hope
to reconstruct it exactly; but it will be worth while to give some
idea, at least, of what it was like. Let us assume that Schlick
used the same size of tempered fifth for each of the six diatonic
fifths; a somewhat larger, but still flat, fifth for the four chro-
matic fifths; and a sharp fifth for the two fifths A^-E^ and C*-
G*. Call these temperaments x, y, and -z respectively. Then,
since the ditonic comma must be absorbed in the course of the
tuning,

6x + 4y - 2z = 24 cents.

Now x is larger than y; let us assume that x = 2y. Since Schlick
said that most of his fifths were to be "somewhat" flat and the
other two fifths "somewhat" sharp, let us assume that x = z.
Then

12y + 4y - 4y = 24 cents, y = 2 cents, x = z = 4 cents.

Thus Schlick' s diatonic fifths, of 698 cents, will be tempered by
1/6 comma; his chromatic fifths, of 700 cents, will be the same
size as those in equal temperament; his two sharp fifths will be
of 706 cents. His diatonic thirds will be six cents sharp; his
chromatic thirds, 8 or 10 cents; the thirds E-G* and A^-C, 18
cents (not unbearable); and the "foreign" thirds, B-D#, F#-A#,
and D^-F, 26 cents, slightly more than a comma.

The deviations for Schlick' s hypothetical temperament are
less than half as large as those for the modified meantone tem-
perament that Tanaka wrongly ascribed to him — the first tem-
perament in this chapter. His is a good system, holding its own
in comparison with systems that were proposed two or three
centuries later. Of the irregular systems discussed in the first

138

IRREGULAR SYSTEMS

section of this chapter, Schlick's is superior to Mersenne's, Ra-
meau's, Hawkes', and Gallimard's.

Even so, Schlick's system is not so good as that of Gram-
mateus, next to be discussed. Therefore we must not assume
that the present reconstruction has erred on the side of Schlick.
As a temperament, it has far greater significance for us than if
it had been the meantone temperament, with two sharp fifths. It
is an indication that in the early sixteenth century organ temper-
ament was nearer to equal temperament than it generally was
for centuries after this time. Schlick's directions have the added
weight that they represent the practice of an actual organist, un-
concerned with mathematics or the theories of the ancient Greeks.

Modifications of Regular Temperaments

In the next main group of irregular temperaments the diatonic
notes are tuned according to one of the well-known regular tem-
peraments and then each tone is divided equally to form the chro-
matic notes. The oldest and best of them was that of Henricus
Grammateus, or Heinrich Schreyber of Erfurt. Grammateus
tuned the diatonic notes of his monochord according to the Pyth-
agorean ratios. But when it came to the black keys, the "minor
semitones," he followed a different procedure. These were
formed by dividing each tone into two equal semitones by the Eu-
clidean method for finding a geometric mean proportional. Gram-
mateus had a figure to illustrate the construction. Perhaps he
obtained this method of halving intervals directly from Euclid.
But he may have owed it to Faber Stapulensis^ (Jacques le
Febvre), who had shown that it was impossible to divide a ton^
numerically into two equal parts, but that the halving of any in-
terval could be accomplished by geometry. At any rate, Ber-
mudo, whose one tuning method was identical with Grammateus',
did depend upon Faber for the method of constructing the mean
proportionals. Faber exerted great influence upon later writers

7"Arithmetica applicirt oder gezogen auff die edel Kunst musica," an appen-
dix to his Ayn new kunstlich Buech (Niirnberg, 1518) a

^Elementa musicalia (Paris, 1496).

139

TUNING AND TEMPERAMENT

Table 114 . Hawkes' Modified 1/5-Comma Temperament

U 7 2 +£ 4 1 „6 1 7 3 2

Names C°C 'D" Eb 5 E~ = F+5 Fr* G" G#~5 A_1 Bb 5 B"1 C°
Cents 0 83 195 303 390 502 586 698 785 893 1005 1088 1200

M.D. 12.7; S.D. 13.0

Table 115. Gallimard's Modified Meantone Temperament, No. 1

Names C C# D D# E F F# G G# A Bb B C

Deca. 0 212 484 744 969 1263 1461 1747 1980 2232 2526 2716 3010

Cents 0 84 193 297 386 504 582 696 789 890 1007 1083 1200

M.D. 13.3; S.D. 14.9

Table 116. Gallimard's Modified Meantone Temperament, No. 2

Names C C# D D# E F F# G G# A Bb B C

Deca. 0 204 484 734 969 1263 1457 1747 1969 2232 2526 2716 3010

Cents 0 81 193 293 386 504 581 696 785 890 1007 1083 1200

M.D. 14.0; S.D. 15.6

Table 117. Schlick's Temperament (Hypothetical)

Names C° C#_1D" Eb+<5 E~~3 Fe F1"1^'^1"^" B^s B" C°
Cents 0 90 196 302 392 502 590 698 796 894 1002 1090 1200

M.D. 8.0; S.D. 8.6

Table 118. Grammateus' Monochord (Pythagorean with Mean Semitones)
Names C° Cr 2 D° D*^ (Eb+2) E° F° fHg° G#_2A° Bb+^ B? C°
Cents 0 102 204 306 408 498 600 702 804 906 1008 1110 1200

M.D. 3.3; S.D. 4.5

140

IRREGULAR SYSTEMS

who attempted to solve the tuning problem. Especially among
mathematical writers who dabbled in this field, Faber's name
was held in something of the same esteem as that of Boethius.

This monochord division of Grammateus is seen to be of a
subtle and theoretical nature. It is equivalent to dividing the
Pythagorean comma equally between the fifths B-F# and B^-F.
As such, it is identical with Marpurg's tuning K. This tuning
may have been used in practice, but hardly by anyone who was ac-
customed, like Schlick, to tune by ear. Note that it was presented
as a method not for fretted instruments, but for organs. Gram-
mateus said in his introduction: "There follows herewith an
amusing reckoning which serves the art of song called music,
and from such reckoning springs the division of the monochord,
from which will then be taken the proportionate length and width
of the organ pipes after the opinion of Pythagoras."

So far as we know, Grammateus was the earliest writer with
a method for finding equal semitones as applied to a tuning sys-
tem. Of course only ten semitones will be equal, the other two
being twelve cents smaller. Probably many men who later spoke
about equal semitones on the lute may have had in mind some
such division, perhaps made by dividing the tones arithmetically
instead of geometrically.

GanassF had a method for obtaining equal semitones on the
lute and viol by linear divisions, using the ratios of just intona-
tion for his basic scale. Although he described his procedure in
more complicated terms, his monochord might have been tuned
as follows: with A the fundamental, form the minor third C with
the ratio 6:5; form F and G as perfect fourth and fifth to C with
the respective ratios 4:3 and 3:2; divide the space between A and
C into three equal parts for B^ and B; divide the space between
C and F into five equal parts for C*, D, Eb, and E; F* will be
half way between F and G, and G* halfway between G and the oc-
tave A. The construction will be even easier if we start with C:
form F and G as perfect fourth and fifth to C; divide the space
between C and F into five equal parts, between F and G into two
equal parts, and between G and the octave C into five equal parts.

9Sylvestro Ganassi, Regola Rubertina. Lettione seconda (1543); ed. Max
Schneider (Leipzig, 1924), Chap. IV.

141

TUNING AND TEMPERAMENT

In the monochord shown in Table 119, the lengths and ratios have
been added according to Ganassi's directions.

Actually, the above monochord does not quite represent Ga-
nassi's ideas. His lute had only eight frets, so that the position
of the notes above F is rather conjectural. However, he placed
a dot where G, the tenth fret, would naturally fall, and it is rea-
sonable to suppose that he would have made a linear division for
the semitones on either side of G. A greater departure from his
ideas lies in ignoring the tempering of the first and second frets:
the second fret is to be placed higher than 8/9 by the width of the
fret, and the first fret higher than 17/18 by half the width of the
freto Similarly the sixth fret is to be placed lower than 17/24 by
the width of the fret. His drawing for the monochord is made
with unusual care (see Figure H). It appears as if the width of
the fret were about 1/2 of 1 percent of the length of the string.
This tempering would make B*3 and B sharper by about half a
comma, and E*3 flatter by the same amount. The first two changes
would not affect the tuning greatly, but the change in the position
of the sixth fret would be harmful. Since Ganassi was not spe-
cific as to the relative length and breadth of the string, we merely
indicate here that he advocated these three tempered values.

Fig. H. Ganassi's Method for Placing Frets on the Lute and Viol
Reproduced by courtesy of the Library of Congress

142

IRREGULAR SYSTEMS

Table 119. Ganassi's Monochord (Just with Mean Semitones)
Lengths 120 114 108 102 96 90 85
Ratios 19/20 18/19 17/18 16/17 15/16 17/18 16/17

Names

C°

D"1

E"1

F°

Cents

182

281

386

498

597

Lengths

Ratios 19/20 18/19 17/18 16/17 15/16
Names G° x A"1 x B"1 C°
Cents 702 790 884 983 1088 1200

M.D. 6.5; S.D. 7.8

Table 120. Reinhard's Monochord (Variant of Ganassi's)

Lengths

56 2/3

53 1/3

50 2/3

42 1/2

Names

C°

D°

E'1

F°

G°

Cents

204

292

386

498

597

702

790

Lengths

Names

A"1

B"1

C°

Cents

884

983

1088

1200

M.D. 6.5; S.D. 7.8

Ratios

Names

C°

Cents

Ratios

Names

G°

Cents

702

Table 121. Malcolm's Monochord (Variant of Ganassi's)

16/17 17/18 18/19 19/20 15/16 16/17 17/18

x D° x E_1 F°

105 204 298 386 498

18/19 19/20 16/17 17/18 15/16

x A"1 x B-1 C°

796 884 989 1088 1200

M.D. 6.5; S.D. 7.8

x
603

143

TUNING AND TEMPERAMENT

Except for the arithmetical divisions, Ganassi's tuning re-
sembles Grammateus' treatment of the Pythagorean tuning, the
difference being that the basic scale here is just intonation. It
also resembles Artusi's treatment of the meantone temperament,
shortly to be described. But even if Ganassi had used the Eu-
clidean method to divide his tones, his monochord(M.D. 6.0; S.D.
7.3) would have been inferior to either of the other two, since the
diatonic just scale varies more greatly from equal temperament
than either the Pythagorean or meantone does. But this is a good
division, and has the tremendous advantage that it is the easiest
of all chromatic monochords to form.

Ganassi's method was discovered independently by Andreas
Reinhard,^ who described the syntonic tuning, and then gave a
table in which the space of each tone, whether major or minor,
is halved to obtain the chromatic note. His table gave string-
lengths only, beginning with 45 for F. Since he used D° instead
of D"1 , his intervals are in a slightly different order from Ga-
nassi's.

Ten years after Reinhard, his tuning method was taken over
by Abraham Bartolus, * the sole difference being that the latter
began with E (48) instead of F (45). Bartolus gave Reinhard as
his source. At first he advocated the method for keyboard in-
struments, and later prescribed it also for fretted instruments
and bells. This general application of a tuning method is some-
thing that is found in very few theorists of Bartolus' period, most
of whom continued to say with Vicentinothat fretted instruments
used equal temperament, and keyboard instruments, the mean-
tone temperament.

In one of the curious dialogs of Printz' s Phrynis Mytilenaeus-^
this same temperament is mentioned. "Charis" describes it and
gives the string-lengths for the C octave, 360 to 180, thus avoid-
ing the fractions that Reinhard had encountered. Very likely
Printz intended this for Reinhard' s tuning, but his perplexing use

^"Monochordum (Leipzig, 1604).

11Musica mathematica: the 2nd part of Heinrich Zeising's Theatri machi-
narum (Altenburg, 1614), pp. 151 f, 165 ff.

12Part 3, Chap. 6.
144

IRREGULAR SYSTEMS

of anagrams effectively conceals Reinhard's name, if it is indeed
hidden there.

Alexander Malcolm** had a division very similar to those of
Ganassi and Reinhard. In fact, it is the inversion of Ganassi's,
with semitones paired in contrary motion. Although Malcolm
said that the tones were to be divided arithmetically, as 16:17:18,
his table of string-lengths (lengths of chords) represents a very
unlikely division, difficult to make. Marpurg, who called the
system ugly, has represented it by a series of increasing num-
bers, as C, C% D are 48, 51, 54. This would mean that Mal-
colm's ratios are to be taken as vibration numbers, improbable
in view of his own terminology for them.

Since Malcolm's scale contains the same ratios for semitones
as Ganassi's and Reinhard's, although in a different order, the
deviation for the three scales will be the same. But his chro-
matic notes are all five or six cents higher than Reinhard's. It
is very probable that Malcolm intended the same division as
Reinhardo Malcolm stated that Thomas Salmon had written about
this scale. But it is often referred to by Malcolm's name alone.
Certainly these well-nigh equal semitones of Ganassi, Reinhard,
Bartolus, Salmon, and Malcolm represent a long-lived (almost
two centuries) and very good way to divide the octave with ease.

Levens' "Sisteme"" also had linear divisions only, but was
far less successful than those just described. His monochord
had integer numbers starting with 48 for C. Ganassi's system
had only five consecutive semitones formed by equal divisions
of a larger interval, but Levens' had seven, from 42 for D to 28
for A. Thus Levens' consecutive semitones vary in size from 85
to 119 cents. Furthermore, his semitone A-B" is very small
(63 cents), with the Archytas ratio, 28:27; whereas his semitone
Bb-Cb, with the ratio 27:25, is more than twice as large (133
cents) . Levens' deviations are as great as for some varieties of
just intonation.

13A Treatise of Musick (Edinburgh, 1721), p. 304.
Abrege' des regies de l'harmonie (Bordeaux, 1743), p. 87.

145

TUNING AND TEMPERAMENT

Since C is 48 in Levens' tuning, the monochord could easily
be constructed with a foot rule. But it would not be so easy to
construct a monochord of indefinite length for this tuning. A
slight change in the values of A and B would greatly simplify the
construction of the monochord, and at the same time would al-
most cut the deviation in half. It would then be formed thus: Di-
vide the entire string into 8 parts, putting D at the first point of
division, F at the second, and AD at the third. Divide the space
between C and D into two parts for C*. Divide the space between
D and F into three parts, for E" and E, and apply EF twice from
F toward A13, for F* and G. Divide the space from A*5 to the
higher C (midpoint of the string) into four equal parts, for A, B",
and B.

The third distinct method of forming equal semitones upon
the lute stems from Giovanni Maria Artusi. ** But, as with Gram-
mateus' division, only ten of the semitones would be equal. In
pointing out the "errors of certain modern composers," Artusi
gave two examples of "intervals false for singing, but good for
playing on the lute." Thus the diminished seventh, C*-B", in the
beginning of Marenzio's madrigal "False Faith," is "false for
voices and for modulation, but not false on the lute and the
chitarone."

On the lute, he continued, "the tone is divided into two equal
semitones." So far Artusi had been speaking very much as had
his predecessors. But he then stated that the tone in question is
not the 9:8 tone, but the mean tone used on the lute and other in-
struments. Later he called the tempered semitone "the just half
of the mean tone." For constructing this temperament he men-
tioned the mesolabium and the Euclidean construction for a mean
proportional, with references to Zarlino and Faber, The meso-
labium would have been useless for this purpose, unless Artusi
had desired complete equal temperament. But Euclid's method
would have served for constructing meantones from just major
thirds, and then for constructing mean semitones from mean tones.

Since Artusi did not give a detailed account of how his tem-
perament was to be formed, we can only surmise that all the di-

l^Seconda parte dell' Artusi overo della imperfettioni della moderna musica
(Venice, 1603), pp. 30 ff.

146

IRREGULAR SYSTEMS

Table 122. Levens' Monochord (Linear Divisions)

Lengths 48 45 42 40 38 36 34 32 30 28 27 25 24

Names C° Db+1 D Eb+1 E F° x G° Ab+1 A Bbo Cb+2 C°

Cents 0 112 231 316 404 498 597 702 814 933 996 1129 1200

M.D. 16.7; S.D. 19.9

Table 123. Levens' Monochord (Altered Form)

Lengths 48 45 42 40 38 36 34 32 30 28 1/2 27 25 1/2 24

Names C° Db+1 D Eb+1 E F° x G° Ab+1 A Bb° B C°

Cents 0 112 231 316 404 498 597 702 814 902 996 1095 1200

M.D. 8.8; S.D. 10.3

Table 124. Artusi's Monochord (Meantone with Mean Semitones)
(Bonded Clavichord Tuning, No. 1)

_ 1 j_l 1 3 5

Names C° x D f x E~ F 5 x G ? x A" x B~4 C°
Cents 0 97 193 290 386 503 600 697 794 890 987 1083 1200

M.D. 5.7; S.D. 7.6

Table 125. Bonded Clavichord Tuning, No. 2

_i _2 .1 _1 _i _5

Names C° x D 3 x E 3 F 6 x G 6 x A 2 x B 6 C°
Cents 0 97 197 294 394 502 599 698 795 895 992 1092 1200

M.D. 2.6; S.D. 3.8

147

TUNING AND TEMPERAMENT

atonic notes were to be tuned as in the ordinary meantone tem-
perament and the chromatic notes by dividing each of the tones
in half. This is the "semi-meantone temperament" mentioned
by Ellis, I® "in which the natural notes C, D, E, F, G, A, B were
tuned in meantone temperament, and the chromatics were inter-
polated at intervals of half a meantone." According to Ellis, it
had been in use on "the old fretted or bonded clavichords." Un-
fortunately, Ellis did not give the source of this information.

If these bonded clavichords had had their notes paired CC*
DD* E FF# GG# AA* B C, a fixed ratio could have existed be-
tween the notes in each pair, so that C#, for example, would always
be 96.5 cents higher than C. Of course, the two diatonic semi-
tones, E-F and B-C, would be about a comma larger, at 117
cents each.

Some writers have said that the bonded clavichords neces-
sarily used the meantone temperament. But nothing would have
prevented the performer from tuning his diatonic tones sharper
than mean tones. Suppose, for example, it had become the fashion
to diminish the fifth by 1/^ comma, as in Bach's day. Then the
bonded clavichord would have had the scale shown in Table 125.

In this tuning the standard deviation is fairly large because
the semitones E-F and B-C have a deviation of eight cents,, If E
and B are made four cents sharper, the mean deviation is un-
changed, but the standard deviation is reduced to 3.0. This much
can be done without changing the ratio of C to C*. But a bonded
clavichord that was constructed at the time Douwes was writing
(1699; see Chapter in) would have had the ratio of this pair of
notes fixed according to the temperament then in use, perhaps
the 1/6-comma meantone system, and the mean-semitone tuning
would then have been even better than in Table 125.

Furthermore, there is no valid reason why the ratio of the
semitones on a single string could not have been -J2, if the bonded
clavichord had been constructed at a time when equal tempera-
ment was widely accepted. The only difficulty is that the free
clavichords were more common then. But it is nonsense to think

^Alexander Ellis, "On the History of Musical Pitch," Journal of the Society
of Arts, XXVIII (1880), 295.

148

IRREGULAR SYSTEMS

that there was any connection between free clavichords and equal
temperament, except where an old clavichord had retained sem-
itonal ratios that belonged to a type of tuning that had been su-
perseded. Even then, as we have shown, the open strings could
have been tuned so that the instrument as a whole would have
varied only slightly from equal temperament.

The only troublesome situation would occur when the bonded
clavichord had its ratios fixed so that, for example, the semitone
between C° and D"^ was not a mean semitone, but C#~^. Re-
member that Artusi was writing about equal semitones on the
lute, not on the clavichord. And other theorists, advocating
meantone temperament for keyboard instruments, made no dis-
tinction between the clavichord, on one hand, and the organ and
harpsichord, on the other. Let us see, in Table 126, what could
be done when the fixed chromatic semitone has only 76 cents, the
diatonic semitone, 117 cents.

Here we assume that C-C#, F-F#,and G-G*are each 76 cents,
and that D-E^andA-B^are each 117 cents. The other seven sem-
itones are free. If we make them all equal, each will have 105.4
cents. That means that D and A are flatter than in the regular
meantone temperament; E, F, G, and B sharper. After this
somewhat eccentric tuning of the diatonic notes, the deviation is
almost half that of the regular meantone temperament, but is
still not quite so good as that of the old Pythagorean tuning, un-
tempered. Therefore on a bonded clavichord that was built for
the complete meantone temperament, even the most scientific
tuning of the free strings would not make a very acceptable tem-
perament. And such clavichords would certainly have delayed
the acceptance of equal temperament.

A corroboration of Artusi' s method of forming equal semi-
tones on the lute came from Ercole Bottrigari.*' He had clas-
sified instruments by their tuning, as Zarlino had done. He went
on to show that the lute cannot play in tune with the cembalo. If
the E string of the lute is tuned in unison with the E of the cem-

*'I1 dcsiderio, ovvero de' concerti di varii stromenti musicali (Venice, 1594);
new ed. by Kathi Meyer (Berlin, 1924).

149

TUNING AND TEMPERAMENT

Table 126. Bonded Clavichord Tuning, No. 3

Names C C* D Eb E F F# G G# A Bb B C
Cents 0 76 181 298 403 509 585 691 767 872 989 1094 1200

M.D. 12.0; S.D. 13.7

balo, the F's will be out of tune, the G's will again be in tune, and
the G^'s out of tune. He explained that, since on the lute the tone
was divided into two equal semitones, and on the cembalo into
two unequal semitones, then the diatonic semitone E-F, with the
ratio of 16:15 tempered, would be higher on the cembalo than on
the lute; but the chromatic semitone G-G* (25:24 tempered) would
be higher on the lute.

This explanation would be true, even if the lute were in equal
temperament. But the interesting question is why the G's were
in tune if the E's were, and vice versa. If the lute were in equal
temperament, it would have no pitches in unison with the cem-
balo save the one that was tuned to a unison to begin with. Now,
Bottrigari was referring to a tuning in which the order of strings
was D, G, C, E, A, D. Of these the E string was called the "me-
zanina," the middle string. On either D string or on the A string,
the 2nd, 3rd, and 5th frets formed a diatonic sequence — A, B,
C, D or D, E, F, G.

Since the position of the frets was the same on all the strings,
the succession on the E string would have been E, F*, G, A.
Therefore, if the diatonic notes on the D and A strings were tuned
in unison with those on the cembalo, as in Artusi's tuning, the
notes E, F% G, and A on the E string will also be in unison. But
E-F on the lute will behalf a mean tone and so will G-G^, whereas
the E-F of the cembalo will be a tempered major semitone and
the G-G* a tempered minor semitone. (F#-G, about which Bot-
trigari said nothing, will be the ordinary major semitone of the
meantone temperament on both instruments, and will be almost
a comma larger than these other semitones on the lute.) This is
the only reasonable explanation of Bottrigari' s statement, and,
since it was made only nine years earlier than Artusi's account,
we may surmise that this method of tuning was in common use
about 1600. We should be careful, therefore, not to assume that

150

IRREGULAR SYSTEMS

every statement about the use of equal semitones on the lute nec-
essarily meant equal temperament, with the ratio of1 {2 for the
semitone.

Temperaments Largely Pythagorean

A great many irregular temperaments are largely Pythagor-
ean, that is, they contain many pure fifths. This is reasonable
enough, since pure fifths are easy to tune and do not depart
greatly from the fifths of equal temperament. As we shall see,
many of these are typical "paper" temperaments, ill adapted
either to tuning by ear or to setting upon a monochord. But first

we shall examine several that used linear divisions only.
1ft
Martin Agricola, ° who was responsible for a good version

of just intonation, showed a monochord for the lute in which the
diatonic notes, like those of Grammateus, were joined by pure
fifths. To divide the tones into diatonic and chromatic semitones,
Agricola applied the old doctrine that the tone is divisible into 9
commas, 5 for the chromatic semitone and 4 for the diatonic.
He tuned a G string, marking off G* as 5/9 the distance from G
to A. That means that G:G#:A as 81:76:72. Thus the diatonic
semitone G*-A had the ratio 19:18, or almost 94 cents, instead
of 256:243 or 90 cents, and the chromatic semitone 110 cents in-
stead of 114.

Agricola formed his A* and C* like the G*. As there were
only seven frets on this string, he did not give values for D#, F,
and F#. But F is of course a major tone below G, and he had
previously shown ED (although he called it "dis") to be a tone be-
low F. But there B^ had been shown to be a tone lower than C,
20 cents flatter than the A* on the other string. These incon-
sistencies are bound to arise when any unequal tuning is used on
a fretted instrument, as Galilei pointed out. For the sake of a
logical construction, let us assume (see Table 127) that each of
the five tones in the octave is divided into 5 + 4 commas. This
may be slightly better than Agricola' s tuning would have been if

18Musica instrumentalis deudsch (4th ed.; Wittenberg, 1545). Reprinted as
Band 20 of Publikation 'alterer praktischer und theoretischer Musikwerke,
1896. The reference here is to page 227 of the latter.

151

TUNING AND TEMPERAMENT

Table 127. Agricola's Pythagorean- Type Monochord

Names C°C#~5D° D#"6 E° F° F^G0 G^"5 A0 A#_1 B° C°
Cents 0 110 204 314 408 498 608 702 812 906 1016 1110 1200

M.D. 8.3; S.D. 8.6

Table 128 WSng Pho's Pythagorean- Type Monochord

Lengths 900 844 800 751 713 668 633 600 563 534 501 475 450

Names C C* D D* E E# F# G G* A A# B C

Cents 0 111 204 313 403 516 609 702 812 904 1014 1107 1200

M.D. 8.9; S.D. 9.0

he had applied it to an entire octave.

This system, if we can call it a system, is appreciably better
than the ordinary Pythagorean tuning. It contains ten pure fifths;
the fifth B-F# is four cents flat (1/6 comma), and A#-F is twenty
cents flat. But none of the credit belongs to the inventor. Agric-
ola, like many another good man, confused geometrical with arith-
metical proportion The old statement about the sizes of semi-
tones is very nearly correct when geometrical magnitudes are
in question, but is less accurate when applied to linear divisions.
Furthermore, it was a happy accident that led him to make his
chromatic notes sharps. If he had divided the tone G-A into G-
A^-Ainthis same manner, his diatonic semitone would have con-
tained 88 cents, the chromatic, 116, thus diverging more widely
from equality than the Pythagorean semitones do. An accidental
improvement is the best we can say for this tuning of Agricola.

Agricola's approximation for the Pythagorean tuning suggests
the monochord of an early Chinese theorist, W&ng Ph5, who lived
toward the end of the tenth century. ^ Perhaps he was familiar
with the excellent temperament of Ho Tchheng-thyen, but, if so,
was too timid to follow his example. Starting with the Pytha-
gorean tuning for the octave 900-450, he has retained the purity

^Maurice Courant, in Encyclopedic de la musique et dictionnaire du conser-
vatoire, Part 1, Vol. I, p. 90.

152

IRREGULAR SYSTEMS

of G and D. He lowered the pitches of all the other notes by add-
ing two units for C#, D*, E, and E#, and one unit for F#, G#, A,
A*, and B. This was too small a correction for most of the notes,
as can be seen from Table 128, which is comparable to that of
Agricola.

John Dowland is another writer whose tuning system, like
those of Ramis, Grammateus, Agricola, and others, had a strong
Pythagorean cast. In his account of "fretting the lute," C, D, F,
G, and A have Pythagorean tuning.20 The chromatic semitone
from C to C# is 33:31, or 108 cents, not far from the Pythago-
rean of 114 cents. The diatonic semitone from D to E^ is 22:21, or
80 cents, considerably flatter than the Pythagorean of 90 cents.
G* and B*3 form pure fifths to C# and E*5 respectively. An unu-
sual feature of the tuning is F* taken as the arithmetical mean be-
tween F and G, and E (!)asthe mean between E^° and F. The value
for E thus obtained, 264:211, is 388 cents, almost a pure third above
C, instead of the expected Pythagorean third. The third D-F#, of
393 cents, is likewise an improvement. Thus the deviation is
somewhat less than that for the Pythagorean tuning, being almost
the same as that of Agricola' s system. There is no B on this
string, but we have made B a pure fifth above E.

The trend of Dowland' s tuning resembles that of Ornithopar-
chus, whose Micrologus was translated into English by Dowland.
Ornithoparchus' division of the monochord was entirely Pytha-
gorean, a ten-note system extending fromA^ to B by pure fifths.
It was natural for Ornithoparchus to advocate the Pythagorean
tuning, since most of his contemporaries had not yet departed
from it„ But a century later, the Pythagorean tuning was becom-
ing somewhat rare. And yet Dowland' s fellow countryman Thomas
Morley, whose precepts have been quoted by everyone who writes
about Elizabethan music, gave only a Pythagorean monochord.

Unusual ratios are a feature of Colonna's tunings also, al-
though he definitely included some ratios that belong to just in-
tonation as well.21 He is noted in the field of multiple division

20Robert Dowland, Variety of Lute-Lessons (London, 1610). "Of Fretting the
Lute" comes under "Other Necessary Observations to Lute-playing by John
Dowland, Bachelor of Music."

^^Fabio Colonna, La sambuca lincea, p. 22.

153

TUNING AND TEMPERAMENT

for having described an instrument, theSambucaLincea, similar
to Vicentino's Archicembalo, upon which the division of the oc-
tave into §1 parts could be accomplished. His mathematical the-
ory of intervals is very ingenious, including superparticular pro-
portions, but also more subtle fractions. He began with certain
well-known consonant ratios: 1:1 (unison), 6:5 (minor third), 5:4
(major third), 4:3 (fourth), 3:2 (fifth), and 5:3 (major sixth). Then
if a string of the monochord is divided to produce a certain in-
terval, the sounding part of the string should produce with the
other part (the Residuo) either one of the above intervals or a
higher octave of it. This means that if any of the above ratios is
called b:a, intervals derived from it have ratios of the form
(2*b + a):2^b. For example, from 1:1 comes 17:16; from 6:5
comes 11:6; from 3:2 comes 25:24. Colonna's two chromatic
monochords are shown in Tables 130 and 131. Each contains
seven pure fifths and several pure thirds. The worst feature of
both monochords is the 55:54 chromatic semitone of 30 cents (as
G-G^ or B^-B) — not much larger than a comma. Almost as bad
is the 12:11 diatonic semitone of 152 cents, as G*-A or B-C. *
The 27:25 diatonic semitone of 134 cents, as F*~ -G or C* -
D° , is not good either, but is a blemish found also in ordinary
just intonation. A redeeming feature of the first monochord is
the division of the 9:8 tone into 17:16 and 18:17 semitones.

Colonna's division of the 10:9 tone into 12:11 and 55:54 "sem-
itones" is reminiscent of the superparticular division of the 10:9
tone that Ptolemy used for his soft chromatic tetrachord, 5/f5 x
14/15 x 27/28, and of the common division of just intonation de-
rived from Didymus' chromatic, 5/6 x 24/25 x 15/16. ^ Other

22Henri Louis Choquel used a 12:11 semitone between A and Bb and a 33:32
semitone between BD and B, in what was otherwise a monochord in ordinary
just intonation. La musique rendue sensible par la me'chanique (New ed.,
Paris, 1762).

2*\A. m, Awraamoff in 1920 devised a tuning for the chromatic octave that out-
does Colonna's. The natural seventh, 8:7, is exploited in this tuning, and
such superparticular near-commatic intervals occur in it as 49:48 (36 cents)
and 64:63 (27 cents)! "Jenseits von Temperierung und Tonalitat," Melos,
Vol. I (1920).

154

IRREGULAR SYSTEMS

Table 129. Dowland's Lute Tuning

Ratios

33:31

9:8

33:28

264:211

4:3

24:17

3:2

Names

Cents

108

204

284

388

498

597

702

99:62

810

Ratios 27:16 99:56 [396:21l] 2:1
Names A Bb x C

Cents 906 986 1090 1200

M.D. 8.2; S.D. 10.1

Table 130. Colonna's Irregular Just Intonation, No. 1

Lengths 50 48 45 [42 6/17] 40 371/2 36

Ratios 24/25 15/16 16/17 17/18 15/16 24/25 25/27

Names C° C¥~2 D"1 [Eb] E"1 F° F*-2

Cents 0 70 182 287 386 498 568

Lengths 331/3 32 8/11 30 28 4/17 26 2/3 25

Ratios 54/55 11/12 16/17 17/18 15/16

Names G° G* A-1 Bb B_1 C°

Cents 702 732 884 989 1088 1200

M.D. 22.0; S.D. 30.3
Table 131. Colonna's Irregular Just Intonation, No. 2

Lengths 1920 2000 2160 2304 2400 2560 2688

Ratios 24/25 25/27 15/16 24/25 15/16 20/21 14/15

Names C° &* "2 D° Eb+1 E_1 F° F*

Cents 0 70 204 316 386 498 618

Lengths 2880 3072 3200 3456 3520 3842

Ratios 15/16 24/25 25/27 54/55 11/12

Names G° Ab+1 A"1 Bb+1 B C°

Cents 702 814 884 1018 1048 1200

M.D. 29.3; S.D. 33.8

155

TUNING AND TEMPERAMENT

possible divisions of the 10:9 tone are 13:12 and 40:39, which is
somewhat better than Colonna's division, and the linear division
19:18 and 20:19, as inGanassi. Divisions of the 9:8 tone include
17:16 and 18:17, as well as 15:14 and 21:20, both of which Co-
lonna used. Other possible superparticular divisions of the 9:8
tone are 13:12 and 27:26; 12:11 and 33:32; 11:10 and 45:44; and
10:9 and 81:80, this last, of course, being the minor tone and
comma.

Divisions of the Ditonic Comma

The Pythagorean-type temperaments in our second group are
more difficult to construct, in that they contain unusual divisions
of the ditonic comma. By ear, these temperaments would have
been almost impossible in many cases, because there are no pure
intervals to check by as in some varieties of the meantone tem-
perament, nor are there even fairly definite tempered intervals,
such as the C E G* C of equal temperament, which also provide
a good check. For the division of the monochord, these temper-
aments could have been set down readily with the aid of loga-
rithms, and they can be expressed in our modern cents with the
greatest of ease. Computers who did not use logarithms were
able to achieve comparable results by a linear division of the
comma, but had less success if they ignored the schisma which
separates the syntonic from the ditonic comma. In most of our
tables we shall assume, for the sake of convenience, that the di-
tonic comma has been given a correct geometric division, and
shall assign cents values to the intervals accordingly.

The leading exponents of this sophisticated sort of comma-
juggling were Werckmeister, Neidhardt, and Marpurg.^4 Each
has expressed the alteration of his fifths and thirds in the 12th
part of a comma, which, strictly, should be the ditonic comma.

^^See Johann George Neidhardt, Gantzlich erschopfte, mathematische Ab-
theilungen des diatonisch-chromatischen, temperirten Canonis Monochordi
(Konigsberg and Leipzig, 1732), pp. 29 (the Fifth-Circles) and 38 (Third-
Circles). See also F.W.Marpurg, Versuch liber die musikalische Temper-
atur, p. 158, for the lettered temperaments A through L. All other refer-
ences will be indicated in footnotes.

156

IRREGULAR SYSTEMS

Since the ditonic comma is approximately 24 cents, this means
that 2 cents will be taken as the unit of tempering. Thus the oc-
tave would contain 600 parts, or thereabouts. This is an inter-
esting forerunner of the cents representation.

In evaluating this group of temperaments, it should be pointed
out that there are two opposing points of view. Since we are
likely to regard most highly those irregular systems that come
closest to equal temperament, there will be in each subclass a
temperament by Mar pur g or Neidhardt that wins the award be-
cause in it the altered fifths are symmetrically arranged among
the entire 12 fifths of the temperament. In these temperaments
all keys are pretty much alike, whether nearer to C major or F*
major.

But the whole intent of having a "circulating" temperament,
of having the octave "well tempered," was to have greater con-
sonance in the keys most used than in those more remote. This
is made very clear in the writings of Werckmeister and Neid-
hardt. We should fail in our duty, therefore, did we not refer at
the end of this chapter to temperaments we have discussed that
satisfy this ideal of graduated dissonance. Both Werckmeister
and Neidhardt had a proper respect for equal temperament also,
but a fanatic like Tempelhof, ^ writing fifty to seventy-five years
later, could say that equal temperament was the worst possible
temperament because one scale must differ from another in its
tuning!

The simplest alteration of the Pythagorean tuning is to divide
the comma into two equal parts. If the altered fifths are consec-
utive, there will be a temperament somewhat like the modifica-
tion of the meantone temperament shown at the beginning of this
chapter. This is Bamberger's tuning,2** except that he has di-
vided the syntonic comma arithmetically between the fifths D-A
and A-E, thus getting a slightly smaller deviation than if he had
divided the ditonic comma (see Table 132).

2**Georg Friedrich Tempelhof, Gedanken iiber die Temperatur des Herrn
Kirnberger (Berlin and Leipzig, 1775), pp. 10, 18.

2°J. P. Kirnberger, Die Kunst des reinen Satzes in der Musik, Part I, p. 13.

157

TUNING AND TEMPERAMENT

Table 132. Kirnberger's Temperament (1/2-Comma)

Ratios

1 256:243

9:8

32:27

5:4

4:3

45:32

3:2

128:81

Names

C°

Dbo

D°

Eb°

E"1

F°

F*-x

G°

Abo

Cents

204

294

386

498

590

702

792

Ratios

270:161 16:9

15:8

2:1

Names

ifl

Bbo

B-1

C°

Cents

895

996

1088

1200

M.D. 9.0;

S.D. 9

Baron von Wiese's second tuning was exactly the same as
Kirnberger's. He was so confirmed a Pythagorean that he called
E~\ F#"\ andB"1 by the respective names Fb°, Gb°, and Cb°,
each of which would have been 2 cents (the schisma) flatter than
the corresponding syntonic value. However, von Wiese's first
temperament^' actually dic^ divide the ditonic comma, making
his F* the mean between Db and B (Table 133). His ratio for
F#, 5760:4073, is an excellent approximation for the square root
of one-half.

Von Wiese's other three temperaments are respectable
enough, for in them the tempered fifths are separated by a minor
or major third. Since the deviation is the same for all three, we
show No. 3 only (Table 134). Von Wiese has indicated it as ex-
tending from B" to D#; but from the construction it extends from
Gb to B, with the fifths Eb-Bb and B-Gb each tempered by half
the ditonic comma. The best arrangement of the tempered fifths
is for them to be separated by a semitone or a tritone. The lat-

Table 133. Von Wiese's Temperament, No. 1 (1/2-Comma)

Names C° Db° D° Eb° E° F° F*~2 G° Ab° A° Bb ° B° C°
Cents 0 90 204 294 408 498 600 702 792 906 996 1110 1200

M.D. 10.0; S.D. 10.8

^Christian LudwigGustav, Baron von Wiese, Klangeintheilungs-, Stimmungs-
und Temperatur-Lehre (Dresden, 1793), p. 9 (No. 1) and p. 12 (No. 3).

158

IRREGULAR SYSTEMS

Table 134. Von Wiese's Temperament, No. 3 (1/2-Comma)

Names C° Db+5 D° Eb+5 E° F° Gb+"2 G° Ab+5 A0 Bb° B° C°
Cents 0 102 204 306 408 498 600 702 804 906 996 1110 1200

M.D. 5.0; S.D. 6.6

ter arrangement occurs in Grammateus' temperament, shown
earlier in this chapter, which is identical with Marpurg's K.
Note that von Wiese's No. 3 is the same as Grammateus' except
for Bb.

Next in order would be temperaments in which the ditonic
comma is divided among three thirds. Charles, Earl Stanhope^o
advocated such a division, but indicated that the syntonic comma
should be divided among the fifths G-D, D-A, and A-E. This left
the schisma, 2 cents, to be divided among the four fifths from Bb
to G*3, the other five fifths being pure. Thus the four black keys
are only one cent sharper than if the tuning were purely Pytha-
gorean. He might better have divided the ditonic comma among
his first three fifths, and not have had the approximate fifths to
worry over. With the ditonic comma divided among three con-
secutive fifths, the mean deviation is 9.0, the standard deviation
9.7. Stanhope's own temperament (Table 135) is slightly better
than this, just as Kirnberger's was better than von Wiese's No.
1, because the former divided the syntonic comma.

Werckmeister^ has shown a temperament in which the comma
is divided into three parts. It is, however, even less satisfactory
than Stanhope's, because it contains five fifths flat by 1/3 comma,
two fifths sharp by 1/3 comma, and only five perfect fifths (see
Table 136) . This is the poorest of the three temperaments Werck-
meister called "correct."

Bendeler has used the 1/3 -comma tempering in two of his

28"Principles of the Science of Tuning Instruments with Fixed Tones," Philo-
sophical Magazine, XXV (1806), 291-312.

29Andreas Werckmeister, Musicalische Temperatur (Frankfort and Leipzig,
1691), Plate.

159

TUNING AND TEMPERAMENT

Table 135. Stanhope's Temperament (1/3-Comma)

Lengths

120

113.84

107.1

101.19

Names

C°

Dbo

D" 3

Ebo

E"1

Cents

197

295

386

498

Lengths

75.89

71.7

67.5

Names

G°

Abo

A""

Bbu

B-1

C°

Cents

702

793

892

996

1088

1200

85.38
nbo

589

M.D. 7.8; S.D. 8.7

three organ temperaments.^ In the first, the tempering is shared
by the fifths C-G, G-D, and B-F* (Table 137) . Since these are
not all consecutive fifths in the circle of fifths, his deviation is
considerably less than Stanhope's.

In Bendeler's second temperament (Table 138), the comma is
divided among the three fifths C-G, D-A, and F*-C#. Since the

Table 136. Werckmeister's Correct Temperament, No. 2 (1/3-Comma)

Names C° C#~3D

f'-'g"

,#■

.b+-i

E- e 3 F° F" 'G3 G" 3A 3 B"'3 B C°
Cents 0 82 196 294 392 498 588 694 786 890 1004 1086 1200

M.D. 9.2; S.D. 10.7

Table 137. Bendeler's Temperament, No. 1 (1/3-Comma)

J-i

Eb° E"

F#_1 G"

G#-iA-3 Bbo

Names C° C" *D 3 E"" E 3 F° F' * G 3 G" *A 3 B"u B 3 C°
Cents 0 90 188 294 392 498 588 694 792 890 996 1094 1200

M.D. 5.0; S.D. 5.8

Table 138. Bendeler's Temperament, No. 2 (1/3-Comma)

Names C° C^D"3 E*30 E

F° Fff 3 G

G A

B" B 3 C°

Cents 0 90 196 294 392 498 596 694 792 890 996 1094 1200

M.D. 4.0; S.D. 4.8

J. P. Bendeler, Organopoeia (2nd ed.; Frankfurt and Leipzig, 1739), p. 40
(No. 1) and p. 42 (No. 2).

160

IRREGULAR SYSTEMS

fifths are more widely separated than before, the deviation is
less than for No. 1.

The best arrangement of the three tempered fifths is to have
them separated by major thirds, as in Marpurg's I, where E and
G* are the same pitches as in equal temperament (see Table 139).

The most famous of Werckmeister's irregular divisions has
the comma divided equally among the four fifths C-G, G-D, D-A,
and B-F#.31 since three of these fifths are consecutive, the de-
viation is comparatively large (see Table 140). This is the only
temperament that Sorge has ascribed to Werckmeister. The
same division was accepted by Marpurg, and a modern acousti-
cian, Karl Erich Schumann, ** has followed suit, without men-
tioning any secondary source.

In Werckmeister's third "correct" temperament (Table 141),
five fifths (D-A, A-E, F#-C#, C#-G#, and F-C) are flattened by
1/4 comma, and one fifth, G*-D#, is raised by the same amount.
Thanks, however, to the more nearly symmetrical arrangement
of the tempered fifths, the deviation is slightly less than for his
first temperamento

In his third temperament, Bendeler,^^ unhampered by a sharp
fifth and with a fairly symmetrical arrangement of the four flat-
tened fifths (C-G, G-D, E-B, G*-D#), succeeded in achieving a
very good division (Table 142).

But, as usual, the best temperament for a particular division
of the comma is completely symmetrical, and so Neidhardt, in
his fourth Fifth-Circle (Table 143), gave Eb, F#, and A the same
pitches they would have in equal temperament. (Marpurg's H is
identical with this.)

When the comma is divided into five parts and the tempered
fifths are arranged as symmetrically as possible, the deviation
begins to approach the vanishing point. (Paradoxically, this de-
viation is lower than for a wholly symmetrical arrangement of
six fifths tempered by 1/6 comma, shown in the next section.) In

31Werckmeister (see Table 1-iO), loc. cit.
32Akustik (Breslau, 1925), p. 31.
33Organopoeia, p. 42.

161

TUNING AND TEMPERAMENT

Table 139. Marpurg's Temperament I (1/3-Comma)

Names C° Cf^D° E°+* E~3 F+3 F*~3 G° G#" A0 Bb+3 B"5 C°
Cents 0 106 204 302 400 506 604 702 800 906 1004 1102 1200

M.D. 3.0; S.D. 3.5

Table 140. Werckmeister's Correct Temperament, No. 1 (1/4-Comma)

Names C° C#_1D~2 £b° E~4 F° F^G'4" G#~l A~4 Bb° B~4 C°
Cents 0 90 192 294 390 498 588 696 792 888 996 1092 1200

M.D. 6.0; S.D. 7.5

Table 141. Werckmeister's Correct Temperament, No. 3 (1/4-Comma)

Names C° C^D0 E 4 E 2 F * ¥w 2G° Gff A 4 B 4 B 2 C°
Cents 0 96 204 300 396 504 600 702 792 900 1002 1098 1200

M.D. 5.0; S.D. 5.7

Table 142. Bendeler's Temperament, No. 3 (1/4-Comma)

Names C° Cff *D 2 E E 2 F° Fff' 4 G 4 G* 4 A 2 Bu B~4 C°
Cents 0 96 192 294 396 498 594 696 798 894 996 1092 1200

M.D. 3.3; S.D. 3.7

Table 143. Neidhardt's Fifth-Circle, No. 4 (1/4-Comma)

U-2 _J h+-1 -i #--! -i H-2 _i ho -i

Names C° C 4D 4 E 4E 2 F° F 2G 4 G 4A 4 B B 2 C°
Cents 0 96 198 300 396 498 600 696 798 900 996 1098 1200

M.D. 2.7; S.D. 2.8

Table 144. Marpurg's Temperament G (1/5- Comma)

Names C° Cff 5D 5 E 5E 5 F° F* 5 G° G * * A" 5 B 5 B~~5 C°
Cents 0 100 199 299 398 498 602 702 802 901 1001 1100 1200

M.D. .7; S.D. 1.3

162

IRREGULAR SYSTEMS

Marpurg'sG (Table 144) this near-symmetrical division is made.
Marpurg called the amount of tempering 2^-/12 = 5/24 comma,
which would be 5 cents, slightly larger than 1/5 comma or 4.8
cents. Although the difference between the two is wholly negli-
gible, the latter amount of tempering has been used in making
the table, with the values rounded off to even cents.

The 1/6-comma temperament is recommended by Thomas
Young, 34 as a simpler method than the irregular temperament
described later in this chapter. In his own words, "In practice,
nearly the same effect may be very simply produced, by tuning
C to F, Bb, Eb, G#, C#, F# six perfect fourths; and C, G, D, A,
E, B, F# six equally imperfect fifths." In other words, he had
six consecutive fifths tempered by l/6ditonic comma (see Table
145). As a practical tuning method, this would not be difficult,

Table 145. Young's Temperament No. 2 (1/6-Comma)

Names C° Dbo D_3 Ebo E~3 F° Gbo G"* Ab° A~2 Bb° B~« C°
Cents 0 90 196 294 392 498 588 698 792 894 996 1090 1200

M.D. 6.0; S.D. 6.8

and it certainly does differentiate between near and remote keys.
This is the tuning of the Out-Of-Tune Piano, the sort of tuning
into which a piano originally in equal temperament might fall if
played upon by a beginner .35 Young's key of G is the best, that
of Db the worst. If he had commenced his set of tempered fifths
with F instead of C, the key of C would have been best.

In Neidhardt's second Fifth-Circle (Table 146), all the fifths

Table 146. Neidhardt's Fifth-Circle, No. 2 (1/6-Comma)

Names C° Cff 2D° E 6 E 3 F « F 3G~? G 3A""e B 3 B "2 C°
Cents 0 102 204 298 400 502 604 698 800 902 1004 1098 1200

M.D. 3.0; S.D. 3.4

34" Outlines of Experiments and Inquiries Respecting Sound and Light," Philo-
sophical Transactions, XC (1800), 145.

35j. Murray Barbour, "Bach and The Art of Temperament," Musical Quar-
terly, XXXm (1947), 66 f, 89.

163

TUNING AND TEMPERAMENT

are altered by 1/6 comma, nine being lowered and three raised.
Since the arrangement is completely symmetrical, the deviation
is low.

Of course, a symmetrical arrangement of fifths alternately
pure and lowered by 1/6 comma comes closest to equal temper-
ament. Both Neidhardt (Third Fifth-Circle) and Marpurg (F)
have presented this temperament (Table 147). Observe that in
it the consecutive notes are alternately the same as in equal tem-
perament and 2 cents higher, so that the mean deviation and
standard deviation both are equal to 2.0. More elaborate patterns
of semitones either 2 cents higher or lower than in equal tem-
perament could be obtained by having two pure fifths alternate

Table 147. Neidhardt's Fifth-Circle, No. 3 (1/6-Comma)

a-i --1 h+i --1 +-1 u-1 u-2 -1 hj- ' -J

Names C° C* 2D 6 E 3E 3 F 6 F 2G° Gff 3A 6 B 6 B 3 C°

Cents 0 102 200 302 400 502 600 702 800 902 1000 1102 1200

M.D. 2.0; S.D. 2.0

with two tempered fifths, or by having three pure fifths similarly
alternate with three tempered ones.

Bermudo,36 wno had also formed equal semitones on the lute
by the method of Grammateus, made a real contribution to tuning
theory in a chapter "concerning the seven-stringed vihuela upon
which all the semitones can be played." This was a method in-
tended for experienced players. His account of the division is
necessarily lengthy and need not be given as a whole. G is the
fundamental, and there are 10 frets, thus making no provision
for F# on this string. The notes from E^ to G inclusive are
formed by a succession of pure fifths. The thirds G-B and A-C*
are each 2/3 syntonic comma sharper than pure thirds. The tone
G-A is 1/6 comma less than a major tone. Then D and E form pure
fourths with A and B, respectively, and G* is a fourth below C*.

The geometry, which consists of linear divisions only, is easy
to follow, especially with the aid of Bermudo's monochord dia-

3" J. Bermudo, Declaracion de instrumentos musicales (Ossuna, 1555), Book
4, Chap. 86.

164

IRREGULAR SYSTEMS

gram (see Figure I). In ratios, as will be seen in Table 148, it
becomes quite complicated, and, if these ratios were to be rep-

r a b C D b f c

b 1 v 1 ■ — !-■-, \ -i-H 1 — i — i i : ; ,

Fig. I. Bermudo's Method for Placing Frets on the Vihuela

Reproduced by courtesy of the Library of Congress

Table 148. Bermudo's Vihuela Temperament (1/6- 1/2-Comma)

Names

G°

G#"

Bb°

c°

Ratios

492075:463684

540:481

32:27

1215:964

4:3

Cents

102.9

200.3

294.2

400.6

498.0

Names

i i

2 D"6

Eb°

F°

G°

Ratios 164025:115921 720:481 128:81 405:241 16:9 [218700:11592l] 2:1
Cents 600.9 698.3 792.1 898.6 996.1 [1098.9J 1200

M.D. 3.9; S.D. 4.2

resented by least integers, as was done in many of these systems,
the fundamental note G would have to be 62,985,600! Let us as-
sume that F*, the unused 11th fret, is a pure fourth above C*.

The reason Bermudo's system is presented in connection with
the use of fifths tempered by 1/6 comma is that that is precisely
what he has. If the temperament of successive fifths is examined,
it will be seen that the fifths on G, A, and B are each tempered
by 1/6 comma, eight fifths are pure, and the usual wolf fifth, G#-
E", is 1/2 comma flat. (It really should not be called a wolf fifth,
since it is flat, not sharp, and the usual poor thirds of the mean-
tone temperaments, on B through G#, are the best of all!)

This is the first time, so far as is known, that any writer had
suggested the formation of notes used in equal temperament by
the proper division of the comma for those notes. Of course he
was making an arithmetical division of the syntonic comma, and
thus had small errors. But so did the late seventeenth and most
of the eighteenth century comma-splitters from Werckmeister
to Kirnberger and Stanhope. Bermudo's three tempered fifths

165

TUNING AND TEMPERAMENT

are as symmetrically arranged as in the Neidhardt-Marpurg sys-
tem shown before this. It is too bad he did not continue his proc-
ess by tempering D# by 2/3 comma and E# by 5/6 comma. Then
he would not have had the half-comma error concentrated on a
single fifth, nor a Pythagorean third on E^. But this method of
Bermudo is worthy of our respect as a very early approach to
equal temperament, somewhat difficult, but not impracticable for
a skilled performer to use.

Werckmeister is the only later writer to temper his fifths by
the 7th part of a comma, perhaps following the example of Zar-
lino's 2/7-comma variety of meantone temperament.^' But his
Septenarium temperament is a rather eccentric thing. In it the
fifths C-G, BO-F, and B-F# are 1/7 comma flat; F#-C# is 2/7
comma flat; G-D is 4/7 comma flat; D-A and G*-D* are 1/7
comma sharp; the remaining five fifths are pure. (The cents

Table 149. Werckmeister's Septenarium Temperament (1/7-Comma)
Lengths 196 186 176 165 156 147 139 131 124 117 110 104 98

5 14 11 5 X it 4 t-» 4

Names C° Cri D~7 ED+"7 E_"7 F° F#_7G~7 G A~~7 B 7 B~7 C°
Cents 0 91 186 298 395 498 595 698 793 893 1000 1097 1200

M.D. 4.7; S.D. 5.6

Table 150. Symmetric Septenarium Temperament (1/7-Comma)

M-4 _J h+ 2 _2 +J U 4 _ 1 U 5 _2 h,J 3

Names C° C" 7D 7 E 7E "7 F 7 Fff""7G 7 Gff_7A 7 B 7 B" 7 C°
Cents 0 100 201 301 401 501 598 699 799 899 999 1100 1200

M.D. 0.5; S.D. 1.0

values have been worked out from Werckmeister' s string-lengths ,
and are slightly inaccurate.)

For the sake of a comparison with Werckmeister's temper-
ament, a symmetric version of the 1/7 -comma temperament is
shown in Table 150. It is even nearer equal temperament than
Marpurg'sG, which had a symmetric distribution of the fifth part
of the comma.

3'A. Werckmeister, Musicalische Temperatur, Plate.
166

IRREGULAR SYSTEMS

Next we have a large group of temperaments in which some
fifths are tempered by 1/5 comma and others by 1/12 comma,
while the remaining fifths are pure. Since 1/12 comma is the
temperament of the fifth of equal temperament, there will be as
many pure fifths as there are fifths tempered by 1/5 comma.
This group of temperaments might be considered, therefore, as
variants of the previously described temperaments in which there
are six pure fifths and six fifths tempered by 1/6 comma.,

Neidhardt was the great inventor of temperaments in which
the comma was divided into both 6 parts and 12 parts. 38 All
three "circulating" temperaments fall into this group. They hap-
pen to be among the poorest ofthis type that he or the other the-
orists have evolved — that is, when compared with equal temper-
ament. But we shall see that they do satisfy Neidhardt' s purpose
in creating them. The first circulating temperament (Table 151)
has four fifths in each group — pure, tempered by 1/12 comma,
and by 1/6 comma. Since four consecutive fifths in it are tem-
pered by 1/6 comma, it may be considered a variant of the 1/5-
comma meantone temperament.

The first of Thomas Young's pair of temperaments is very
like the Neidhardt temperament shown in Table 151. ^ Young
said, "It appears to me, that every purpose maybe answered, by
making C:E too sharp by a quarter of a comma, which will not
offend the nicest ear; E:G# and At>:C equal; F*:A* too sharp by
a comma; and the major thirds of all the intermediate keys more
or less perfect as they approach more or less to C in the order
of modulation."

Table 151. Neidhardt's Circulating Temperament, No. 1
(1/12-, 1/6-Comma)

_ J h° - -3

A 2 B B 4 C°

894 996 1092 1200

mes C° C#""6D~"3 Eb°

-- #--5 --1 U-
E a F° Fff 6 G 6 G*

nts 0 94 196 296

392 498 592 698 796

M.D. 4.0; S.D. 4.6

"*°J. G. Neidhardt, Sectio canonis harmonici, pp. 16-18.
39Thomas Young, in Philosophical Transactions, XC (1800), 145 f.

167

TUNING AND TEMPERAMENT

Young accomplished the first result by tempering the fifths
on C, G, D, and A by 3/16 syntonic comma, and the other results
by tempering the fifths on F, B", E, and B by approximately 1/12
syntonic comma, and leaving the other four fifths pure. The total
amount of tempering would be 13/12 syntonic comma, this being
sufficiently close to the ratio of the ditonic to the syntonic comma.
Young has given numbers for his monochord, and they agree well
with his theory. He has made a mistake, however, in calculating
the length for ED (83810), which was intended as a pure fourth be-
low gC The corrected length is given in Table 152.

Table

152. Young's Temperament

No. 1

(1/12- , 3/1C

-Comma

)

Lengths

Names

100000
C°

94723

C 12

89304

3
D «

84197

79752

e""4

74921

Y+ i2

71041

#--
Y i2

Cents

196

298

392

500

592

Lengths
Names

66822

G 16

63148
G 12

59676

A"16

56131

53224

5
B"6

50000
C°

Cents

698

796

894

1000

1092

1200

M.D. 5.3; S.D. 5.9

Now 3/16 syntonic comma is an awkward interval to deal with.
If, instead, we take 1/6 ditonic comma as the temperament of
Young's four diatonic fifths, and 1/12 ditonic comma for his sec-
ond group of fifths, his monochord will be precisely of theNeid-
hardt type. The differences from the monochord he did give are
so small that the cents values do not differ. The arrangement of
his second group of fifths is slightly different from Neidhardt's,
and this accounts for the difference in deviation.

Mercadier's temperament (Table 153) closely resembles
Young's, even to the total amount of tempering — 13/12 syntonic
comma. ^ He directed that the fifths from C to E should be flat
by 1/6 syntonic comma, and those from E to G* flat by 1/12 comma.
Then G# is taken as AD, the next three fifths are to be just, and
the fifth F-C then turns out to be about 1/12 comma flat.

40Antoine Suremain-Missery, Theorie acoustico-musicale (Paris, 1793),
p. 256.

168

IRREGULAR SYSTEMS

Table 153. Mercadier's Temperament (1/12-, 1/6-Comma)

Names C°C* 12D 3 E 12E 3 F 12 Fff 6 G 6 G* J A "2 B 12B~4 C°
Cents 0 94 197 296 394 500 594 698 794 895 998 1094 1200

M.D. 4.1; S.D. 4.5

Table 154. Marpurg's Temperament D (1/12-, 1/6-Comma)

#_ 2 _i h+i --£- '{- J ! # 3 1 l, l 1

Names C° C 3D 4E 4E 12 F° F 2 G 8 G 4A~4 B ~2B" C°
Cents 0 98 198 300 398 498 600 698 798 900 998 1098 1200

M.D. 1.3; S.D. 1.6

Table 155. Neidhardt's Circulating Temperament, No. 2
(1/12-, 1/6-Comma)

u-2 _i h+- -- +- #-^ --i ¥-- -- b+- --

Names C° Cff 4D 3 E 6E 12 F 12 F 3G 6 G 6A 2 B 6 B 12 C°
Cents 0 96 196 298 394 500 596 698 796 894 1000 1096 1200

M.D. 3.3; S.D. 3.7

Table 156. Neidhardt's Circulating Temperament, No. 3
(1/12-, 1/6-Comma)

#_3 _i h+- -— #--2 --1 #--5 --1 b+-i- --

Names C° C 4D 3E 6E 12 F° Fff 3 G 4 Gff 6 A 2 B 12 B 12 C°

Cents 0 96 196 298 394 498 596 696 796 894 998 1096 1200

M.D. 2.7; S.D. 2.9

Table 157. Neidhardt's Third-Circle, No. 4 (1/12-, 1/6-Comma)

Names C° Cff 4D 3 E E 2 F° Fff 3G 6 Gff 6A 2 B 6 B 3 C°
Cents 0 96 196 296 396 498 596 698 796 894 1000 1094 1200,

M.D. 2.7; S.D. 3.4

169

TUNING AND TEMPERAMENT

As usual, Marpurg has presented the symmetric version (Ta-
ble 154) of the above temperaments. It has negligible deviations.

In the second and third of Neidhardt's "circulating" temper-
aments, six fifths are tempered by 1/12 comma, and three each
are pure or are tempered by 1/6 comma. These two tempera-
ments (Tables 155 and 156) are quite similar, both containing
three consecutive fifths tempered by 1/6 comma. Thus they pos-
sibly represent the extreme case of modification of the 1/6-comma
meantone temperament. Number 3 has a shade greater sym-
metry and hence smaller deviation.

Temperaments 4 and 3 of Neidhardt's Third-Circle have de-
viations very similar to those of the temperaments shown in Ta-
bles 155 and 156. In fact, their mean deviations are equal re-
spectively to those of No. 2 and No. 3 in these tables, but their
standard deviations are higher because they contain some sharp
fifths. In No. 4 (Table 157), there are three fifths tempered by
1/12 comma and five by 1/6 comma; three fifths are pure, and
one is 1/12 comma sharp. In No. 3 (Table 158), four fifths are
1/12 comma flat, six are 1/6 comma flat, and two are 1/6 comma
sharp. (The same tempered fifths as in No. 3 appear in our hy-
pothetical version of Schlick's temperament, but differently ar-
ranged.)

Once again Marpurg has given the symmetric version of Neid-
hardt's temperaments, specifically of the second and third "cir-
culating" temperaments.

Logically we show next two temperaments (Tables 160 and
161) in which eight fifths are flat by 1/12 comma and two by 1/6
comma, while two are pure. Such a temperament is the fifth of
Neidhardt's Third-Circle.

The temperament shown in Table 160 comes so close to equal
temperament that in practice it could not be improved upon. But
the canny Marpurg has halved its deviation by using greater sym-
metry (see Table 161).

Another temperament of Neidhardt has the same deviations as
those of his fifth Third-Circle (Table 160). This is the fifth tem-
perament in his Fifth-Circle (Table 162), in which six fifths are

170

IRREGULAR SYSTEMS

Table 158. Neidhardt's Third-Circle, No. 3 (1/12-, 1/6-Comma)

Names C° C"*D""^ E^ e"^¥+T2 ¥f~T2G~~6 G*""6A"T2 Bb+* b" C°
Cents 0 96 196 296 394 500 598 698 796 896 1002 1092 1200

MD. 3.3; S.D. 4.7

Table 159. Marpurg's Temperament C (1/12-, 1/6-Comma)
Names C C#"3D--s Ebn E"i Fo FMG-*GHA-* Bb+° B~& C°
Cents 0 98 200 300 400 498 600 700 800 898 1000 1100 1200

M.D. 1.0; S.D. 1.4

Table 160. Neidhardt's Third-Circle, No. 5 (1/12-, 1/6-Comma)

U — 1 h+ 1 — 4- 1 U-— -— U 2 -- ho-1 -1

Names C° C ff~12D~« E * E »F • Fff 12G 12Gff~~3A 4 B " B_ 2 C°
Cents 0 100 200 300 398 502 598 700 800 900 1000 1098 1200

M.D. 1.3; S.D. 2.0

Table 161. Marpurg's Temperament B (1/12- , 1/6-Comma)

■# 2 i hx1 1 -uJ- #-i -i #-.3 -i h + i - —

Names C° C 3D~4 E° +4 E 3 F+12Fff 2G 6 G* 4A 3 B° 6 B 12 C°
Cents 0 98 198 298 400 500 600 698 798 898 1000 1100 1200

M.D. .7; S.D. 1.1

Table 162. Neidhardt's Fifth-Circle, No. 5 (1/12- , 1/6-Comma)

Names C° Cff 12D 6 E 6E * F 6 F* 2G 12G# 3A 3 B 4 B~3 C°
Cents 0 100 200 298 402 502 600 700 800 898 1002 1102 1200

M.D. 1.3; S.D. 2.0

171

TUNING AND TEMPERAMENT

flat by 1/12 comma and four by 1/6 comma, while two are sharp
by 1/12 comma.

The remaining temperaments in this group come from Mar-
purg. The first (Table 163) of his temperaments in which some
fifths are sharp contains six fifths flat by 1/6 comma, and three
fifths each flat or sharp by 1/12 comma. ***■ Obviously, this is a
variant upon the temperament in which six fifths are flat by 1/6
comma, the other six pure. The mean deviation, 2.0, is the
same, but, as expected, the standard deviation is higher here.
Other possible variants would contain, in addition to the six fifths
tempered by 1/6 comma, two fifths each flat or sharp by 1/12
comma or pure; or four pure fifths and one each flat or sharp
by 1/12 comma.

The second temperament (Table 164) in this other set byMar-
purg has fifths that do not differ greatly from those in the pre-
vious temperament. Here the six fifths are tempered by the un-
usual amount of 5/24 comma (shown as the same fraction that
did duty as 1/5 in his Temperament G, but really 5/24 this time),
and three each are pure or 1/12 comma sharp.

In Marpurg's Temperament A (Table 165), ten fifths are flat
by 1/12 comma, and one each is pure or 1/6 comma flat. This is
as far as one can go in this direction, for the next step would be
to have twelve fifths flat by 1/12 comma — that is, equal tem-
perament.

The other limit for this sequence of temperaments by Mar-
purg is his own Temperament F, already shown as Neidhardt's
Fifth-Circle, No. 3 (Table 147). In it there are no fifths tem-
pered by 1/12 comma, and six fifths each pure or flat by 1/6
comma. Just before it in the set comes Temperament E (Ta-
ble 166), which has two fifths flat by 1/12 comma, and five fifths
each pure or flat by 1/6 comma.

Marpurg's Temperament E, shown in Table 166, has the least
deviation of the five temperaments in the set. Note the devia-
tions again: A, 1.7,1.8; B, 0.7,1.1; C, 1.0, 1.4; D, 1.3, 1.6; E,
0.3,0.8. From the table for E it is easy to see why its deviation
is low: there are seven consecutive notes with cents values end-

^Marpurg, Versuch liber die musikalische Temperatur, p. 163.
172

IRREGULAR SYSTEMS

Table 163. Marpurg's Temperament, No. 1 (1/12- , 1/6-Comma)

Names C° C 2D 12 E 12E 3 F 6 F 12G 12G 3A 6 B 4 B 4 C°
Cents 0 102 202 304 400 502 602 704 800 902 1002 1104 1200

M.D. 2.0; S.D. 2.4

Table 164. Marpurg's Temperament, No. 2 (1/12-, 5/24-Comma)

a_3 _± b+i _i __1 #_3 __5 #_2 __5. b_JL _i?

Names C° C# 4D 12E 8 E 3 F 12 Fw 4G 24 G 3A 12 B 12B 24 C°
Cents 0 96 194 297 400 496 594 697 800 896 994 1097 1200

M.D. 3.0; S.D. 3.1

Table 165. Marpurg's Temperament A (1/12- , 1/6-Comma)

Names C° C# 2 D 6 E 4E 4 F 12 Fff 12G 12Gff 12A 6 B • B 3 C°
Cents 0 102 200 300 402 500 602 700 802 902 1000 1102 1200

M.D. 1.7; S.D. 1.8

Table 166. Marpurg's Temperament E (1/12-, 1/6-Comma)

Names C° C#_T2D" 12Eb+-5E~4 F+1 F#"^G_^G#~ 3 A-^ B°+* B_" C°
Cents 0 100 202 302 402 502 602 700 800 900 1000 1100 1200

M.D. 0.3; S.n0.8

Table 167. Neidhardt's Fifth-Circle, No. 6 (1/12-, 1/4-Comma)

M-— _J K+i -- i U -i -_L #5,1 hj^ 1 7

Names C° Cff 12D 3 E 4E 3 F 12 F 2G 12G 6A 4 B 6 B-" C°
Cents 0 100 196 300 400 496 600 700 796 900 1000 1096 1200

M.D. 2.7; S.D. 3.3

Table 168. Neidhardt's Fifth-Circle, No. 9 (1/12-, 1/4-Comma)

ji.i _i h+i -i ik 2 J. a 2 > hn ±

Names C° C# ^ 3E 4 E 3 F° Fff 3 G 12 G 3A"3 B B" 12 C°

Cents 0 98 196 300 400 498 596 700 800 898 996 1100 1200

M.D. 2.0; S.D. 2.4

173

TUNING AND TEMPERAMENT

ing in 00, and five ending in 02. Therefore the total deviation
will be only 4 cents, or a mean deviation of 0.3. In the other
temperaments of the set, some values end in 00 and others in 98
or 02. But in no other temperament do all the 00' s come together
as they do in E. Therefore the deviation is higher in the other
temperamentSo But it need not have been higher. If in A the pure
fifth is followed directly by the fifth flat by 1/6 comma, there
will be only one note with an 02 ending, and eleven notes with 00.
The fifths in B, C, and D can be so arranged that there will be
respectively 2, 3, and 4 consecutive notes with an 02 (or 98) end-
ing, the other endings being 00. Thus the minimum deviation
(M.D. 0.3; S.D. 0.8) will be the same for all five temperaments,
but this will not always involve the most symmetrical version of
the fifths.

The remaining nine temperaments are all by Neidhardt, and
each contains some fifths tempered by 1/4 comma. His Fifth-
Circle, No. 6 (Table 167) has four fifths each flat by 1/4 comma
or flat or sharp by 1/12 comma. His arrangement is symmetric.

In Temperament No. 9 of this same set (Table 168), Neidhardt
has three fifths flat by 1/4 comma, three flat by 1/12 comma, and
six pure. Again the arrangement is symmetric. The deviation
is lower than for the previous temperament.

In Temperaments 7 and 10 (Table 169 and 170), Neidhardt di-
vides the comma into 4 or 6 parts. No. 7 is especially compli-
cated, having eight fifths flat by 1/6 comma and two sharp by 1/6
comma, and one each flat or sharp by 1/4 comma. It would be
difficult to construct a symmetric arrangement from such an ar-
ray, and Neidhardt has not attempted to do so.

Table 169. Neidhardt's Fifth-Circle, No. 7 (1/6-, 1/4-Comma)

Names C° C# 6D 12E 6 E 3 F 6 F* 3 G 4 G* 3A 12B 3 2 C°

Cents 0 94 194 298 400 494 596 696 800 892 996 1098 1200

M.D. 3.3; S.D. 4.1

174

IRREGULAR SYSTEMS

Table 170. Neidhardt's Fifth-Circle, No. 10 (1/6-, 1/4-Comma)

U-- -- h-l-i -2 a 2 1 i, 5 1 Un 2

Names C° C* 6D 4 E 6 E 3 F° F f 3G * G « A~~2 B B~3 C°
Cents 0 94 198 298 392 498 596 696 796 894 996 1094 1200

M.D. 3.0; S.D. 3.8

Table 171. Neidhardt's Fifth-Circle, No. 10, Idealized

ji_J _i h+i --Z- u-J. _i u-- --5. hn _-l

Names C° C# 6D 4 E 6 E 12 F° F* 12G 4 G* 6 A 12B B 12 C°
Cents 0 94 198 298 398 498 598 696 796 896 996 1096 1200

M.D. 1.3; S.D. 2.4

Table 172. Neidhardt's Sample Temperament, No. 2
(1/12-, 1/6-, 1/4-Comma)

Names C° Cw D l2 E E 12 F 12 F 12G 6 G A 3 B 12 B 12 C°
Cents 0 90 194 294 386 496 590 698 792 890 994 1088 1200

M.D. 6.3; S.D. 7.2

Temperament 10 (Table 170) is considerably simpler, with two
fifths flat by 1/4 comma, three by 1/6 comma, and the remaining
seven pure. The deviation is slightly lower than for No. 7.

But in No. 10 also the arrangement is far from symmetric.
Let us see what would result from an approach to symmetry. Al-
though the deviation is about halved in Table 171, it is possible
that, as in the alphabetically named temperaments by Marpurg,
the least deviation for all four of these Neidhardt temperaments
will not occur with the most nearly symmetric arrangement of
the fifths.

In the remaining five temperaments in this group, Neidhardt
has tempered his fifths by 1/4, 1/5, and 1/12 comma. His second
and third "sample" temperaments (the first was just intonation)
have relatively high deviations. 42 No. 2 (Table 172) has three
fifths flat by 1/4 comma, one by 1/5, two by 1/12, five pure, and
one 1/12 comma sharp.

J. Go Neidhardt, Gantzlich erschopfte mathematische Abtheilung, p. 34.

175

TUNING AND TEMPERAMENT

Neidhardt's No. 3 (Table 173) is somewhat less erratic than
No. 2, with six pure fifths, and two each flat by 1/4, 1/6, or 1/12
comma. It also has a lower deviation than No. 2.

Rather similar to the above sample temperaments is his
Third-Circle, No. 1 (Table 174), in which five fifths are pure,
two flat by 1/4 comma, one by 1/6, and four by 1/2.

Two temperaments from the Fifth-Circle are considerably
better than the three just mentioned. In No. 11 (Table 175) there
are no pure fifths; two fifths are flat by 1/4 comma, two by 1/6,
five by 1/12, while three are 1/12 comma sharp.

Table 173. Neidhardt's Sample Temperament, No. 3
(1/12-, 1/6-, 1/4-Comma)

» U _i ho— i. 5 U 2 x H H -1 ho 5

Names C°Cff"12D 3 E 12E-'5 F° Fff "6 G"5 Gff " 12A" 12B B 6 C°

Cents 0 92 196 296 388 498 592 698 794 892 996 1090 1200

M.D. 5.7; S.D. 6.4

Table 174. Neidhardt's Third-Circle, No. 1 (1/12-, 1/6-, 1/4-Comma)

Names C° C 6D 4 E 12E 4 F° F 8G 12 G 12A 2 B 12 B 4 C°
Cents 0 94 198 296 390 498 592 700 794 894 998 1092 1200

M.D. 5.3; S.D. 5.9

Table 175. Neidhardt's Fifth-Circle, No. 11 (1/12- , 1/6-, 1/4-Comma)

u-2 -J h+— -— +— 4--1 -X u-3 _J ho --

Names C° C* 4D 4 E 12E 12 F 12 Fff 12G 12Gff 3A 2 B B 2 C°

Cents 0 96 198 296 394 500 598 700 800 894 996 1098 1200

M.D. 2.7; S.D. 3.2

Table 176. Neidhardt's Fifth-Circle, No. 12 (1/12- , 1/6- , 1/4-Comma)

Names C° C^ 12D~4 E 4 E_2 F° Fff"2 G" 12G 4 A~4 B B"2 C°

Cents 0 100 198 300 396 498 600 700 798 900 996 1098 1200

M.D. 2.0; S.D. 2.3

176

IRREGULAR SYSTEMS

In No. 12 (Table 176) there are six pure fifths, and two each
flat by 1/4, 1/6, or 1/12 comma. This has precisely the same
number of each size of fifth as the third sample temperament,
in which the deviation was almost three times as great. The
reason, of course, is to be found in the symmetry of No. 12.

Metius' System

At the beginning of this chapter it was said that "by making
the bounds sufficiently elastic" all irregular systems could be
classified. That statement is severely tested by the final tuning
method listed in this part of the chapter, one presented by Ad-
rian Metius. It was not possible to see Metius' own description,
and Nierop, who gave the monochord, seemed to have been puz-
zled by it himself. * Nierop has shown this monochord in two
forms, one from 1000 to 500 and the other from 11520 to 5760,
with E the fundamental. It is evident from the context that the
second monochord was given simply to show how its lengths have
been increased or diminished by arithmetic divisions of the syn-
tonic comma, and that only the first table comes from Metius
directly.,

By using Metius' lengths, it is possible to reconstruct the
tempering, indicated by the exponents. Apparently there is only
one pure fifth, C-G. The fifths on BD and A are 1/12 comma flat,
those on F and E 1/6 comma flat, on B and C* 1/2 comma flat,
and on G 3/4 comma flat! The fifths on D and F* are 1/6 comma
sharp, that on D* 1/3 comma sharp, and on G* 1/2 comma sharp.

Metius' system does not seem to follow any known system of
temperament or modification thereof. Specifically, it does not
resemble the meantone temperament, for only the thirds on B*3
and E are pure, the other thirds varying in size up to 417 cents
for GD-BD and 419 cents for AD-C. But there is no pattern ap-
parent in the alterations, no planned shift from good to poor keys.
The fifth G-D, 3/4 comma flat, is almost as unsatisfactory as
this same fifth would be in just intonation. There is no good rea-
son for both of the fifths B-F# and C#-G# to be half a comma flat

4**D,, r. van Nierop, Wis-konstige Musyka (Amsterdam, 1650). The reference
here is to page 60 of the 2nd edition (1659).

177

TUNING AND TEMPERAMENT

and then to have the fifth G#-D# half a comma sharp. All in all,
Metius has been just about as erratic as he could be.

And yet the system, despite its irregularities, is much bet-
ter than the ordinary 1/4-comma meantone temperament and is
slightly better than the Pythagorean or the 1/6 -comma mean-
tone. That much we must grudgingly admit. Metius' tempera-
ment contains eight different sizes of fifth. But that is not much
less regular than many of the fairly good temperaments we have
shown that had four sizes of fifth, while Werckmeister's Septe-
nariumand Neidhardt's second sample temperament had five dif-
ferent sizes. And so let us label it highly irregular, but not
really unworkable.

"Good" Temperaments

With Metius' enigmatic temperament we have described the
last of our irregular tuning systems, and are in a position to try
to formulate a judgment upon them. It is easy to see how the
modifications of the Pythagorean, just, or meantone system by
the halving of tones, as in the systems of Grammateus, Ganassi,
or Artusi, would make these systems much more like equal tem-
perament. But it is more difficult to see what Werckmeister,
Neidhardt, and Marpurg were driving at in their multifarious at-
tempts to distribute the comma unequally among the twelve fifths.

If, as was pointed out at the beginning of an earlier section
of this chapter, our ideal is equal temperament, we shall praise
highly some of the beautifully symmetric systems of Marpurg
and Neidhardt. But the trouble is that they are too good! The
deviations for most of them are lower than for a piano allegedly
tuned in equal temperament by the most skillful tuner. In some
cases these temperaments might have been successfully trans-
ferred from paper to practice by calculating the number of beats
for each of the beating fifths. Since most of the fifths were to be
tuned pure, such a method might have been easier than that pur-
sued today. These same temperaments might have been reduced
to distances on a monochord with slightly greater ease than equal
temperament could be, although it must be remembered that us-
ually even the most innocent set of cents values needs logarith-

178

IRREGULAR SYSTEMS

mic computation before yielding figures for a monochord. But it
will be safe to dismiss most of these oversubtle systems as use-
less, even for the age when they were devised.

What do we have left? It will be of interest to consider which
of his twenty systems Neidhardt considered the best. In the Sec-
tio canonis he had said, "In my opinion, the first [of the circu-
lating temperaments] is, for the most part, suitable for a vil-
lage, the second for a town, the third for a city, and the fourth
for the court." The fourth was equal temperament; the mean
deviations of the other temperaments had been 4.0, 3.3, and 2.7
cents, respectively.

In the much later Mathematische Abtheilungen Neidhardt pre-
sented eighteen different irregular temperaments, together with
just intonation and equal temperament. He then attempted to
choose the best of these twenty tunings. He chose equal temper-
ament, of course, and the two temperaments (Third-Circle, No.
2, and Fifth-Circle, No. 8) that were identical with the first and
second circulating temperaments above. Now half of the rejected
temperaments had deviations lower than that of the second cir-
culating temperament (3.3), and a couple of others were just
about as good. But none of these was considered worthy in the
final appraisal, Neidhardt had, incidentally, changed his ideas
somewhat as to the relative position of the best temperaments:
the Circulating Temperament, No. 2 (Fifth-Circle, No. 8) is now
considered best for a large city; No. 1 (Third-Circle, No, 2) for
a small city; and Third-Circle, No. 1, not included before, for a
village.

If we examine the deviations of the major thirds in the three
temperaments Neidhardt himself considered superior, we quickly
find why he liked them. In the second circulating temperament
(Table 155) the thirds on Cand Fare 8 cents sharper than a pure
third, and the sharpness gradually increases in both directions
around the circle of fifths until the three worst thirds are 18
cents sharp. In the first circulating temperament (Table 151)
the third on C is only 6 cents sharp, and there is the same grad-
ual increase until the five poorest thirds are all 18 cents sharp.
In the Third-Circle, No. 1 (Table 174), the third on C is 4 cents
sharp, and the six poorest thirds are either 18 or 20 cents sharp.

179

TUNING AND TEMPERAMENT.

Werckmeister' s third temperament, the first of the three he
has labeled "correct" (Table 140), is much like the Neidhardt
temperament just mentioned. Its thirds on C and F are only 4
cents sharp, but the thirds of the principal triads in the key of
D*3 are all a syntonic comma, 22 cents, sharp. Werckmeister
himself said that some people who advocated equal temperament
held that "in the future ... it will be just the same to play an air
in D*3 as in C. ^ But he held consistently "that one should let
the diatonic thirds be somewhat purer than the others that are
seldom used." 45

A good comparison can be made between two temperaments
of Neidhardt, already mentioned as having fifths of four different
sizes and the same number of each size, but with a different ar-
rangement. The Fifth-Circle, No. 12 (Table 176) has a sym-
metric arrangement and a low mean deviation, 2.0. Its thirds
show no trend whatever from near to far keys, but are sufficiently
irregular to make this seem a poor attempt at equal tempera-
ment. Not so its companion, the third sample temperament (Ta-
ble 173), in which the third on C is only 2 cents sharp, whereas
four of the five poorest thirds are 20 cents sharp. To be sure,
the deviation for this temperament, 5.7, is almost three times
as great as for the other one, and there is a painful lack of sym-
metry. But the unsymmetric temperament is "circulating," and
therefore deserves an honored place among the "good" temper-
aments,,

Thomas Young's temperaments also deserve mention for their
circulating nature. His first temperament (Table 152) is equiva-
lent to a temperament with four pure fifths and four fifths each
tempered by 1/6 or 1/12 comma. It is constructed with scien-
tific accuracy so that the thirds range in sharpness from 6 cents
forC-E to 22 cents, a syntonic comma, forF#-A#. Its mean de-
viation is 5.3. On the other hand, there is the symmetric form
of this temperament, Marpurg's D (Table 154), with a mean de-
viation of 1.3. And the even better, nonsymmetric form, with a

"A. Werckmeister, Hypomnemata musica (Quedlinburg, 1697), p. 36.

45Werckmeister, Musicalische Paradoxal-Discourse (Quedlinburg, 1707),
p. 113.

180

IRREGULAR SYSTEMS

mean deviation of 0.3! But these last-mentioned temperaments
are curiosities only, whereas Young's differentiated admirably
between near and far keys.

However, Young's first temperament was too difficult to con-
struct, as he had described it with fifths tempered by 3/16 and
"approximately" 1/12 syntonic comma. Therefore he substituted
his second method (Table 145), which was of the utmost simplic-
ity, with six consecutive perfect fifths and six consecutive fifths
tempered by 1/6 ditonic comma. Its mean deviation was 6.0. In
it the thirds on C, G, and Dare each 6 cents sharp, whereas those
on F% C#, and G* are each 22 cents sharp. Neidhardt's Fifth-
Circle, No. 3 (Marpurg's F) is the symmetric version of this
temperament (Table 147), with a mean deviation of 2.0. Again
we may well say that Young's version is an excellent irregular
temperament, while the symmetrical version represents having
fun with figures.

So many versions of good circulating temperaments have ap-
peared on these pages, each with its points of excellence, that we
cannot resist the temptation to close this chapter with an irreg-
ular temperament to end irregular temperaments! Gallimard's
modification of the ordinary meantone temperament, by a sys-
tematic variation in the size of the chromatic fifths, was good
enough in principle, but could not have been too successful be-
cause of the large number of other fifths tempered by 1/4 comma.

What is really needed, in order to have a more orderly change
in the size of the thirds, is to have the variable tempering ap-
plied to all the fifths, instead of to only five of them. Let the
fifth D- A be the flattest, and let each succeeding fifth in both di-
rections around the circle of fifths be a little sharper until the
fifth on A^ is the sharpest. Then the total parts to be added will
be 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36 parts. Since these
parts are to be added to 12 fifths, it is evident that D-A, the flat-
test fifth, will be flatter than the fifth of equal temperament by
three of these parts; the fifths B-F# and F-C will be precisely
the size of the equal fifth; and the sharpest fifth, AD-E°, will be
larger than the equal fifth by three parts. The thirds will vary
as follows (the error being expressed as the number of parts be-
low or above the third of equal temperament): C-E, -8; G-B, -8;

181

TUNING AND TEMPERAMENT

D-F#, -6; A-C#, -2: E-G#, 2; B-D#, 6; Gb-Bb, 8; Db-F, 8;
Ab-C, 6; Eb-G, 2; Bb-D, -2; F-A, -6.

We can choose the value for one part that will give the de-
sired size of thirds. If the part is one cent, the fifth D-A is 697
cents, practically a meantone fifth, and the fifth Ab-Eb is 703,
practically perfect; the best thirds, C-E and G-B, are 392, 1/4
comma sharp; the poorest thirds, Gb-Bb and Db-F, are 408,
precisely a Pythagorean third.

Table 178 should have satisfied the desire of Werckmeister
and his contemporaries for a circulating temperament in which
all the thirds are sharp, but none more than a comma, and all
the fifths are flat or pure. As the size of the part is reduced,
the tuning approaches equal temperament. When the part is in-
creased to 1 3/4 cents, the best thirds are pure. But the poor-
est thirds are now 414 cents, about 5/4 comma sharp. Thus Ta-
ble 178 probably represents the limit of a tolerable temperament
in the extreme keys. Since the mean deviation for the entire se-
ries of temperaments formed in this manner is precisely pro-
portional to the size of the part, it would be easy to devise a sys-
tem with the deviation of any of the systems in this chapter, but
with a more orderly distribution of the errors, as regards com-
mon keys and less-used keys.

The Temperament by Regularly Varied Fifths may be re-
garded as the ideal form of Werckmeister' s "correct" temper-
aments and of Neidhardt's "circulating" temperaments and of all
"good" temperaments that practical tuners have devised by rule
of thumb. Let us see, therefore, how closely it is approached by
these other temperaments. In Table 179, the deviations have
been computed, not only from equal temperament, but also from
our temperament with variable fifths. The table shows clearly
that the temperaments with greatest symmetry do not fit so well
into the desired pattern as do those that are much less regular
in their construction. In general, the temperaments with lowest
deviation from the one ideal temperament will have a high devi-
ation from the other. Neidhardt's second circulating tempera-
ment has the unique position of ranking the same with regard to
both.

182

IRREGULAR SYSTEMS

Table 177. Metius' Irregular Temperament

Lengths 1000 940 896 837 800 749 704 668 628 596 563 530 500

"3 G#_1 A+~& Bb+T5B
M.D. 9.5; S.D. 11.6

Names E° F 6 F# 3 G 3 Gff A 12 B 12B 6 C 3 C# 2D 12D#~2 E°

Table 178. Temperament by Regularly Varied Fifths

Names CxDxEFxGxAxB C
Cents 0 92 197 297 392 500 591 699 794 894 999 1091 1200

M.D. 5.8; S.D. 6.6

Table 179. Deviations of Certain Temperaments

From

Equal

Temperament

From Varied Fifths

M.D.

S.D.

M.D.

S.D.

Neidhardt's Circulating, No. 1

4.0

4.6

2.1

2.3

No. 2

3.3

3.7

3.3

3.7

No. 3

2.7

2.9

4.2

4.7

Third- Circle, No. 1

5.3

5.9

1.2

1.5

Wer ckmeister 's

Correct, No. 1

6.0

7.5

1.9

2.3

No. 2

9.2

10.7

4.7

5.7

No. 3

5.0

5.7

3.8

4.2

Neidhardt's Fifth-Circle, No. 12

2.0

2.3

6.2

6.7

Sample, No. 3

5.7

6.4

1.5

1.8

Young's No. 1

5.3

5.9

1.7

1.9

Marpurg's Letter D

1.3

1.6

6.7

7.1

Young's No. 2

6.0

6.8

1.9

2.0

Neidhardt's Fifth-Circle, No. 3

2.0

5.0

5.8

Schlick's (Hypothetical)

8.0

8.6

2.7

3.1

Neidhardt's

Third- Circle, No. 3

3.3

4.7

3.0

3.8

Our hypothetical reconstruction of Arnold Schlick's tempera-
ment had the same size of fifths as Neidhardt's Third-Circle, No.
3, but differently arranged, and with a fairly high deviation. Ob-
serve that, with this other standard of varied dissonance, Schlick's
temperament is even a little better than Neidhardt's. Of all the
temperaments shown in our table, Neidhardt's Third-Circle, No.

183

TUNING AND TEMPERAMENT

1 seems to be the best, with our new standard, although Neid-
hardt himself said it was best for a village! But it would have
been difficult to tune, and therefore Thomas Young's Tempera-
ment, No. 2 probably cannot be surpassed from the practical
point of view. Even so, the highest honor must be paid to old
Arnold Schlick, writing so long before these other men, but stat-
ing as clearly as need be for his very practical purpose, "Al-
though they will all be too high, it is necessary to make the three
thirds C-E, F-A, and G-B better, „ . .as much as the said thirds
are better, so much will G be worse to E and B."

Table 180. Compass of the Lute

G Tuning

A Tuning

3 4 5 6

2 3 4 5

7 8

Bb B C Db(C#)

B C C#(Db) D

E F

F F# G Ab(G#)

F# G G#(Ab) A

B C

C C#D Eb(D#)

C# D D#(Eb) E

F# G

Ab A Bb Cb(B)

A BbB (Cb) C

D Eb

Eb E F Gb(F#)

E F F#(Gb) G

A Bb

Bb B C Db(C#)

B C C#(Db) D

E F

184

Chapter VIII. FROM THEORY TO PRACTICE

In our intensive study of scores of tuning systems we have failed
to note what may be learned from the music itself. Some of the
theorists who have written on tuning were able composers as well.
When they described with precision a particular division of the
monochord, their theory may well have coincided with fact. But
the tuning theories of the mere mathematicians do not carry so
much weight. Nor do the rules of thumb the musicians more
commonly presented. All of these theories may be put into neat
little pigeonholes, but one can be sure that the practice itself,
because of the limitations of the human ear, was even more var-
ied than the extremely varied theories.

It is not to be expected that a study of the music will provide
a precise picture of tuning practice. It is to be used more by
way of corroborating what the theorists have said. Let us con-
sider first the contention of Vicentino that the fretted instruments
were always in equal temperament. In general we can reach
certain conclusions concerning tuning by examining the range of
modulation. However, this is not definitive as regards the lutes
and viols. Korte listed D#'s in lute music from 1508, an A* from
1523, and many D^'s from 1529.* But the mere presence of
notes beyond the usual 12 -note compass proves little, because
the lutes were not restricted to a total compass of 12 semitones.
As shown in Table 180, the normal compass with the G tuning
was Ct> to C# and for the A tuning from DD to D#.

Ordinarily, lutes and viols had six strings, tuned by fourths,
with a major third in the middle. Thus the open strings might
be G C F A D G or A D G B E A. It is easy to see here the
prototype of Sch'dnberg's chords built by fourths. Because of the
perfect fourths, the fretted instruments might have inclined to-
ward the Pythagorean tuning, as the later violins have done.
Mersenne pointed out that the major third in the middle would

iOskar Korte, "Laute und Lautenmusik bis zur Mitte des 16. Jahrhunderts,"
Internationale Musikgesellschaft, Beiheft 3 (1901).

TUNING AND TEMPERAMENT

then be sharp by a comma.

But the strings of lutes and viols were tuned by forming uni-
sons, fifths, or octaves with the proper frets on other strings,
thus making the tuning uniform throughout the instrument. Vin-
cenzo Galilei^ stated that if the tuning were not equal, semitones
on the A string (mezzana) of the lute based on G would have the
note names shown in Table 180. Since the frets were merely pieces
of gut tied straight across the fingerboards at the correct places,
the order of diatonic and chromatic semitones would have to be
the same on all strings. Thus the chromatic compass of a lute
with six strings and eight frets would be as shown in Table 180,
if meantone temperament had been used.

There might be some question for the G tuning regarding
notes produced by the 6th fret, since B would be a better choice
than C" on the 4th string. But the remaining notes for the 6th
fret agree somewhat better with other notes in the compass than
the equivalent sharped notes would have done. Galilei pointed
out that Gt> (4th string, 1st fret) was not a pure fifth to C# (3rd
string, 4th fret), nor was DD (5th string, 1st fret) a pure octave
to the C#. He might have added that DD (1st string, 6th fret) was
not a pure octave above the C# either.

It is easy to multiply examples of unsatisfactory intervals on
the unequally tuned lute in G. (Read them a tone higher for the
A tuning.) Try building major triads upon the notes of the 6th
string, starting with B13. C, D*3, and E*3 are satisfactory as roots
also, but false triads are generated on B and D. On the 5th
string, starting with D, the satisfactory triads are on E13, F, and
A13; false triads on D, E, Gb, and G. On the 4th string, starting
with G, the only unsatisfactory triad is on C13. On the 3rd string,
starting with C, the other satisfactory triads are on D, E", and
F, with false triads on C* and E. Thus, of 26 major triads in
close position, only 17, about 2/3, are available. Some of the
triads, those on G, D, and A, unsatisfactory in the lower octave,
can be played correctly in the higher octave. But the complete
E and B major triads are unavailable anywhere, because there
are no G* and D# — unless, of course, the 6th fret runs to sharps
rather than to flats.

^Fronimo (Venice 1581; revised edition, 1584), pp. 103 f.
186

FROM THEORY TO PRACTICE

As illustrations of incongruous notes on particular frets, let
us examine some of the Austrian lute music of the sixteenth cen-
tury, as found in Volume 18 of the Austrian Denkm'aler. The
first collection represented is Hans Judenkiinig's Ain schone
kunstliche Underweisung (1523), His third Priamell is modal,
but often suggests C minor. Like most of the German and Aus-
trian composers, Judenkiinig used the A tuning of the lute. In
bar 3 the note aD appears as the 4th fret on the 2nd string, indi-
cating that this fret has a flat tuning (see Table 180). But in bar
4 there is a b and in bar 19 a c# , both of which belong to the
sharp tuning for this fret.

For Judenkiinig's fourth Priamell the editor has put the sig-
nature of three sharps, as an indication of the prevailing sharp-
ness. This even extends to the 6th fret, which would then include
an e* . Actually there is an e# in the music, and no f '. There-
fore it would have been possible to play this piece with an une-
qual temperament, but not without changing the 6th fret from its
normal flat tuning,

Simon Gintzler's fifth Recercar (1547) used the Italian G tun-
ing. Here the 6th fret has a flat tuning, as shown by aD and a
very frequent eb . But in bar 10 there is a b instead of the c*5
belonging to the flat tuning. In Gintzler's setting of Senfl's song
"Vita in ligno moritur," the 6th fret is again flat, but in bar 15
both ab and b occur.

The a*3 and b also occur several times in Bakf ark's Fantas-
ias (1565). More interesting is his setting of "Veni in hortum
meum, soror mea" (1573). In bar 50, d* ' occurs as the third of
the B major triad, indicating a sharp tuning for the 6th fret. This
means that f' is not available on this fret; but f ' does occur in
bar 56 and elsewhere. In bar 62 the complete C minor triad oc-
curs: c' eb g' c", with the eb' the 4th fret on the 3rd string.
But this fret must have had a sharp tuning, since the notes d#'
g , and c occur on it with great frequency.

It would be easy to multiply examples, from the music of Ital-
ian, French, and Spanish composers. Those that have been given
are sufficient to show that in the golden age of lute music the
composers were indifferent to discords that would have arisen

187

TUNING AND TEMPERAMENT

if an unequal temperament had been used. The example from
Judenklinig occurs so early in the century (1523) that it seems
very probable that lutes and viols did employ equal temperament
from an early time, perhaps from the beginning of the sixteenth
century.

We need not be too much concerned with what the equal tem-
perament for the fretted instruments was really like. It might
have been the Grammateus-Bermudo tuning — Pythagorean with
mean semitones for the chromatic notes* It might have been the
Ganassi-Reinhard mean semitones applied to just intonation, or
Artusi's more subtle system of mean semitones in meantone
temperament. Or the frets might have been placed according to
Galilei's 18:17 ratio, or (correctly) according to Salinas' ratio
of the 12th root of 2. In any case, it would have been a good,
workable temperament.

Tuning of Keyboard Instruments

In the early sixteenth century Schlick and Grammateus de-
scribed systems for keyboard instruments that came close to
equal temperament, and the correct application of Lanfranco's
tuning rules must have resulted in equal temperament itself.
But these systems were anomalous for a day when few acciden-
tals were written. Examples of organ music from the late fif-
teenth and the entire sixteenth century are found in numerous
collections, such as Schering's Alte Meister aus der Friihzeit
desOrgelspiels; Volume 1 of Bonnet's Historical Organ Recitals;
Kinkeldey's Orgel und Clavier in der Musik des 16. Jahrhun-
derts; Volume 1 of Margaret Glyn's Early English Organ Music;
Volume 3 of Torchi's L'arte musicale in Italia; Wasielewski's
Geschichte der Instrumentalmusik im 16. Jahrhundert; Volume
6 of the Italian Classics series.

With the exception of the English composers, the compass
used by all these composers was less than 12 notes — ED-F* or
Bb-C . Both Tallis and Redford had D# in one piece and Eb in
another, thus posing a problem with regard to the tuning. But
except for them, there was no problem about performance: all

188

FROM THEORY TO PRACTICE

of this organ music could have been played on an instrument in
meantone temperament.

Even 12 of Schlick's 14 little pieces (Monatshefte fur Musik-
geschichte, 1869) lie within a compass of Eb-C#. One of the re-
maining pieces has an A^; the other, G#. Since Schlick had di-
rected that the wolf be divided equally between the fifths C*-G*
and Ab-ED, these notes would have caused him no difficulty.
Perhaps Tallis and Redford were dividing the error similarly.

Much the same can be said for the clavier music of this pe-
riod. Merian's Per Tanz in den deutschen Tabulaturbuchern
(Leipzig, 1927) contains about 200 tiny keyboard pieces, and Vol-
ume 2 of Bbhme's Geschichte des Tanzes about 20 more. None
exceeds the Eb-G# compass. The famous English collection of
virginal music, Parthenia, reveals nothing beyond the fact that
Byrd preferred"!^ the younger composers Bull and Gibbons,
D*. In Margaret Glyn's edition of Gibbons' Complete Keyboard
Works, five of the 33 virginal pieces have a D#, but only two con-
tain Eb's, one of these, a Pavan in G minor, having also an AD.
But that does not necessarily mean that Gibbons did not use the
meantone temperament. The virginals could have been set for
an A13 at one time and for a D* at another — a point that will be
discussed at some length later. More significant are the A^ and
D# that occur in a G minor Fancy for organ by Gibbons. Unless
Gibbons' tuning was appreciably better than the meantone tem-
perament, this Fancy would have had some very rough places.
This same Ab-D# was used in Tarquinio Merula's Sonata Cro-
matica, a work having a modern ring because of its chromati-
cism.3

Just a word about chromaticism. Other things being equal,
a piece that contains many chromatic progressions is more likely
to have an excessive tonal compass than one that is not chro-
matic. But, since there are 12 different pitch names in the mean-
tone compass, Eb-G#, it is entirely possible for a chromatic
piece to lie within it. A Toccata by Michelangelo Rossi, for ex-
ample, published in 1657, is very chromatic, but carefully re-

3Luigi Torchi, L/arte musicale in Italia (Milan, post 1897), III, 345-352.

189

TUNING AND TEMPERAMENT

mains within the meantone bounds.4

The great English manuscript source of the early seventeenth
century, the Fitzwilliam Virginal Book, is a monument to the
boldness of the clavier composers of that time. Naylor^ has
given a fascinating and exhaustive account of the music in this
collection, and has shown that many of the progressions contain-
ing accidentals resemble modulations to our major and minor
keys more than they do modal cadences. Twenty-five of the 297
compositions contain D*'s, with Bull, Byrd, Farnaby, and Tom-
kins in the lead0 Bull, Farnaby, Tisdall, and Oystermayre have
A#'s also.

With one exception, the largest compass in the entire collec-
tion is that of Byrd's "Ut, re, mi, fa, sol, la," which extends
from AD to D*0 That exception, of course, is John Bull's com-
position on the hexachord, with the same title as Byrd's. It over-
laps the circle of fifths by six notes, with the compass CD-A*.
Bull states his Canto Fermo first on G and rises by tones through
A, B, Db, Eb, and F. He then begins afresh with Ab, Bb, C, D,
E, F% and G. An enharmonic modulation occurs at the begin-
ning of Section 4, where the chord of F# is quitted as GD. The
editors of the Fitzwilliam Virginal Book were so impressed with
this passage that they correctly stated in a footnote, "This inter-
esting experiment in enharmonic modulation is thus tentatively
expressed in the MS.; the passage proves that some kind of
'equal temperament' must have been employed at this date." 6

This remarkable composition is not a mere juggling with
sounds, as Nay lor has alleged. It has real musical interest, and
because of its sustained style seems better adapted to the organ
than to the clavier. But do not try to build up a theory of the use
of equal temperament in England during Queen Elizabeth's reign
on the basis of Dr. Bull's composition. Remember that it stands
practically alone. It seems almost as if Bull had written a Fancy
for four viols, and then, led by some mad whim, had transcribed

4 Ibid., p. 309.

^An Elizabethan Virginal Book (London, 1905).

"J. A. Fuller-Maitland and W. Barclay Squire, The Fitzwilliam Virginal Book
(Leipzig, 1899), I, 183.

190

FROM THEORY TO PRACTICE

it for virginals and tuned his instrument to suit.

One of the boldest of the keyboard composers of the early
seventeenth century was Frescobaldi, an exact contemporary of
Gibbons. Of his 31 works for organ and clavier, • three contain
DD, three a D#, and one an A*. One of the most interesting of
these is the Partite sopra Passacagli for organ, with a compass
of D^-G*. The G* is the third of the dominant triad of A minor,
and the D*3 the third of the subdominant triad of F minor. Hence
the ordinary meantone temperament would be inadequate for
Frescobaldi.

In decided contrast to Frescobaldi are Sweelinck (German
Denkm'aler, IV Band, 1. Folge) and Scheidt (German Denkmaler,
I Band). Sweelinck' s Fantasia Cromatica, with E°-D# compass,
was the only one of 36 pieces examined to exceed 12 scale de-
grees, and Scheidt, although not averse to chromaticism and
rather fond of D*'s, had no single composition, of 44 examined,
with more than 12 degrees.

As we reach the middle of the seventeenth century, we shall
have to differentiate more carefully between music for organ
and for clavier. The organ had a fixed compass, usually Eb-G#,
but perhaps B^-D* or AD-C^. Even if the composer did not em-
ploy AD and D#, for example, in the same composition, as Gib-
bons and Merula had done, the presence of these notes in sepa-
rate compositions was an indication that he was using at least a
modified version of the meantone temperament."

Not so for clavier. A study of the accidentals in clavier
music suggests that tuning practice must have accommodated it-
self to the music to be played. The performer would retune when
changing from sharp to flat keys. Bach could tune his entire
harpsichord in fifteen minutes; to change the pitches of only a
couple of notes in each octave would have taken a much shorter
time. Moreover, all the movements of the common dance suites

7I classici della musica italiana (Milan, 1919), Vol. XII; Torchi, op. cit.,
Vol. III.

°The course of the argument and most of the examples in the remaining part
of this section have been taken freely from my article "Bach and The Art of
Temperament," Musical Quarterly, XXXIII (1947), 64-89.

191

TUNING AND TEMPERAMENT

were in the same key, and this helped to restrict the compass to
not more than twelve different pitch names, even if that compass
was not the conventional ED-G^.

The theorists give us little information about the variable
tuning of claviers. Mersenne hinted at the practice^, He had
given two keyboards in just intonation, the first with sharps only
(except for BD) and the second with flats. Current practice, he
said, was represented by either of these, but with tempered, not
just, intervals^ Some eighty-five years later Kuhnau wrote to
Mattheson that the strings of his Pantalonisches Cimbal (a large
keyed dulcimer) vibrated so long he could not use equal temper-
ament upon it, but had to "correct one key or another" when turn-
ing from flats to sharps.

More valuable evidence of the variable tuning practice for
clavier comes from the music itself. Of Froberger's 67 clavier
compositions (Austrian Denkmaler, VI, 2. Theil, and X, 2. Theil),
6 use 14 scale degrees, 10 use 13, and the remaining 51 use 12
or fewer. But only half (26) of the 51 lie wholly within the usual
meantone compass. His accidentals range altogether from G^
to E#.

Similarly, Johann Pachelbel's clavier music (Bavarian Denk-
maler, 2. Jahrgang, 1. Band) suggests a variable tuning. Of 49
compositions examined, only 2 have more than 12 scale degrees.
But of the remaining 47, only 21, or less than half, lie within the
E^-G* compass, and the total range is from D*3 to B*. An ex-
ception among Pachelbel's works, the Suite in AD (Suite ex Gis),
beginning with anAllemand inAD minor, contains an enharmonic
modulation at the point where the Fb major triad is treated as E
major by resolving upon A minor, just before a cadence in E*3
major! With a range from D*30 to B for this single movement,
it seems evident that for the moment Pachelbel was as reckless
as Bach.

Kuhnau' s works (German Denkmaler, IV Band, 1. Folge) give
musical evidence of variability to buttress what he wrote to Mat-
theson. Of his 28 clavier works, 3 of the 6 Biblical Sonatas have
a compass of 14 scale degrees; the other 3 sonatas and 5 other
works have 13. But of the remaining 17 works that have no more

192

FROM THEORY TO PRACTICE

than 12 different pitches in the octave, only 2 lie wholly within
the Eb-G* tuning. Actually Kuhnau preferred equal temperament
upon the clavier. But most of these works would have been pass-
able in meantone temperament if he had "corrected" some of the
notes, just as he did on the Pantalon.

Of Frangois Couperin's 27 charming suites for clavecin, only
6 have no more than 12 different scale degrees. They are all in
the minor key, and in each the flattest note is a semitone higher
than the keynote, as No. 8 in B minor has the compass C-E*.
Twenty of the remaining 21 suites exceed the circle of fifths by
one or two notes. But here again it is characteristic to have the
flattest note a semitone above the tonic. For example, all five
suites in D major-minor have the precise compass Eb-A#. Cou-
perin leaves a strong impression that the dissonance inevitable
in the slightly extended compass was a coolly calculated risk,
and that a variable meantone tuning was used for these suites
also. The one exception is No. 25, in Eb major and C major-
minor. The compass here is 15 scale degrees, from Gb to D*.
This would, perhaps, be carrying piquancy too far.

There is ample evidence that in Italy during the first half of
the eighteenth century equal temperament or its equivalent was
being practiced. Three composers represented in the Italian
Classics had, in a particular composition, a similar compass,
15 notes in the overlapping circle of fifths. They are Zipoli,
Db-D#, Vol. 36; Serini, Cb-C#, Vol. 29; and Durante, Gb-G#,
Vol. 11. Of 70 of Domenico Scarlatti's delightful little "sona-
tas,"9 45, or more than half, overlap the circle. In one sonata
he had a compass of 18 degrees, Db-B#; in another, 17, Gb-A#.
All of these men upon occasion wrote notes so remote from the
tonal center that meantone temperament seems wholly out of the
question. Both Serini and Durante used Fx, and Scarlatti, Cx.

At this time, in Germany, Telemann was advocating a form
of multiple division with 55 notes in the octave, for a clavier
with only 12 notes in the octave, which was practically the same
as Silbermann's 1/6-comma variety of meantone temperament.
We might expect, therefore, that his compositions for clavier

9Heinrich Barth, Klavierwerke von Domenico Scarlatti (4 vols.; Vienna, c.
1901). ~

193

TUNING AND TEMPERAMENT

would not exceed the bounds of the meantone temperament. How-
ever, Telemann's 36 Clavier Fantasies have a total range of G"-
B#, the same as for Couperin's suites. Only 8 of the fantasies
overlap the circle, by one or two degrees. Of the remaining 28,
only one lies within the ordinary meantone bounds, E"-G*. The
others swing to the sharp side or the flat side, depending upon
the key. Thus Telemann undoubtedly used the meantone temper-
ament, but with variable intonation.

It has been suggested in the preceding pages that composers
such as Bull, Gibbons, Frescobaldi, and Domenico Scarlatti,
whose works exceed the meantone bounds by several scale de-
grees, were not using the meantone temperament. Were they,
then, using equal temperament? That question is difficult to an-
swer, especially since there was a type of tuning that would have
been fairly satisfactory in many of these cases. The title of
Bach's great collection of preludes and fugues, Das wohltem-
perirte Clavier, has usually been taken to mean, as Parry called
it, "The Clavichord Tuned in Equal Temperament." But even in
Bach's day there was a good German phrase for equal tempera-
ment — "die gleichschwebende Temperatur," "the equally beat-
ing temperament." Bach's title might better be paraphrased,
"The Well-Tuned Piano."

Now, "well -tuned" had been used in a somewhat technical
sense by the Flemish mathematician Simon Stevin, over a cen-
tury before the first volume of the "48" was compiled in 1722,
and by Bach's great French contemporary Rameau also, with a
meaning nearly the same as Parry has given to it. To German
theorists, however, there was a distinction. Andreas Werck-
meister has erroneously been hailed as the father of equal tem-
perament because of the title of one of his works on tuning,
Musicalische Temperatur, and because of Mattheson's eulogy.
Mattheson had said, "And thus the fame previously divided between
Werckmeister and Neidhardt remains ineradicable — that they
brought temperament to the point where all keys could be played
without offense to the ear."^ (Underscoring is the present au-

10J. Mattheson. Critica musica. II, 162,
194

FROM THEORY TO PRACTICE

thor's.) Werckmeister himself has used the phrase "wohl tem-
perirt" as follows: "But if we have a well-tuned clavier, we can
play both the major and minor modes on every note and transpose
them at will. To one who is familiar with the entire range of keys,
this affords variety upon the clavier and falls upon the ear very
pleasantly."

What did Werckmeister mean by these words? To use Neid-
hardt's phrase, he meant a "completely circulating genus," that
is, a tuning in which one could circumnavigate the circle of fifths
without mal de son, Both men, as we have seen in Chapter VII,
presented a number of different monochords, with the "foreign"
thirds beating as much as a comma. Werckmeister said of them,
"It would be very easy to let the thirds Db-F, Gb-Bb, Ab-C beat
less than a full comma; but since thereby the other, more fre-
quently used thirds obtain too much, it is better that the latter
should remain purer, and the harshness be placed upon those
that are used the least." Elsewhere Werckmeister described
equal temperament with fair accuracy, but demurred, "I have
hitherto not been able to approve this idea, because I would rather
have the diatonic keys purer." And so to Werckmeister "well-
tuned" meant "playable in all keys —but better in the keys more
frequently used."

If, then, a composer exceeded twelve different pitch names
rarely and then only by a few scale degrees, his works could have
been played to good advantage on a "well-tuned clavier." Com-
posers like Bull and Pachelbel and Scarlatti, however, who ef-
fected enharmonic modulations and used double sharps, would
have been badly served even by Werckmeister' s best-known
"correct" temperament, in which the key of Db had Pythagorean
thirds for all its major triads. Equal temperament was needed
for their works. -^

An equal temperament was needed for the keyboard works of]
Bach, both for clavier and for organ. It is generally agreed that
Bach tuned the clavier equally. Actually he was opposed to equal
temperament, in the sense that there must be strict mathemati-
cal ratios, which are first applied to the monochord and from
there to the instrument to be tuned. Of course he was right. The
best way to tune in equal temperament, as Ellis stated, is to

195

TUNING AND TEMPERAMENT

count beats. Have you ever heard of a contemporary piano tuner
who carried a monochordwith him? And yet the underlying the-
ory must be correct or the result will be unsatisfactory: Ellis
could not have given his practical tuning rule with assurance had
he not been able to calculate accurately how far its use would
fall short of the perfection implied by the term "equal tempera-
ment."

The organ works of Bach show as great a range of modulation
as his clavier works do. Except for a dozen chorale preludes in
the Orgelbiichlein, there are only 3 organ works of 148 examined
that do not overstep the compass of the conventionally tuned or-
gan. The compass of individual organ pieces is very frequently
13, 14, and 15 scale degrees, and even 18, 19, and 21 degrees
have been observed. The compass of Bach's organ works as a
whole is E°k-Cx, 25 degrees! In these works is a host of ex-
amples of triads in remote keys that would have been dreadfully
dissonant in any sort of tuning except equal temperament. For
corroboration, if corroboration be necessary, we need but note
the advice that Sorge gave to the instrument-maker Silbermann,
two years before Bach's death. Sorge, a proponent of equal tem-
perament, said: "In a word — Silbermann' s way of tempering
cannot exist with modern practice. I call upon all impartial and
experienced musicians — especially the world-famous Herr Bach
in Leipzig — to witness that this is all the absolute truth. It is

to be desired, therefore, that the excellent man [Silbermann]

should alter his opinion regarding temperament. . . . *

Just Intonation in Choral Music

We have seen that just intonation exists in many different
forms, and that the best version, if modulations are to be made
to keys beyond B^ and A, comes near the Pythagorean tuning, as
with Ramis. The contention has often been made that unaccom-
panied voices sing in just intonation. Zarlino-^ listed instru-

11Georg Andreas Sorge, Gesprach zwischen einem Musico theoretico und
einem Studioso musices, p. 21.

12Sopplimenti musicali, Chaps. 33-37.
196

FROM THEORY TO PRACTICE

merits in three groups, each with a different tuning: keyboard
instruments in meantone temperament; fretted instruments in
equal temperament; voices, violins, and trombones in just in-
tonation. His argument was that since intonation is free for these
three last-named groups, they would use an intonation in which
thirds and sixths are pure. Three hundred and forty-eight years
later Lindsay Norden said, "As we shall show, no singer can sing
a cappella in any temperament .... A cappella music, therefore,
is always sung in just or untempered intonation. ^

Let us see what is implied by these statements. In the first
place, singers must be able to sing the thirds and sixths purely.^
This may sound like a self-evident truth, too absurd to discuss.
But scientific studies of intonation preferences show that the hu-
man ear has no predilection for just intervals, not even the pure
major third. ° Alexander Ellis declared that it was unreliable
to tune the pure major thirds of meantone temperament directly,
preferring results obtained by beating fifths. Hence the singers
must be highly trained to be able to sing the primary triads of a
key justly.

In the second place, the singers must be able to differentiate
intervals differing by the syntonic comma, 1/9 tone. We have
seen that inPtolemy's version of the syntonic tuning theDminor
triad, the supertonic triad of the key of C major, will be false.
If, as Kornerup and others advocate, the Didymus tuning is used
instead of Ptolemy's, the dominant triad will be false, which is
a greater loss„ But a singer trained to niceties of intonation
would have to vary his pitch by a comma in such critical places,
and thus save the situation. Very good. But studies at the Uni-

13n. Lindsay Norden, "A New Theory of Untempered Music," Musical Quar-
terly, XXII (1936), 218.

14Except for the reference to the Italian madrigalists, the remaining part of
this section has been freely adapted from my article "Just Intonation Con-
futed," Music and Letters, XIX (1938), 48-60 by permission of the editor of
Music and Letters, 18 Great Marlborough Street, London, W. 1.

15Paul C. Greene, "Violin Intonation," Journal of the Acoustical Society of
America, IX (1937-38), 43-44; Arnold M. Small, "Present-Day Preferences
for Certain Melodic Intervals . . . ," Ibid., X (1938-39), 256; James F. Nick-
erson, "Intonation of Solo and Ensemble Performance . . . ," Ibid., XXI (1949),
593-95.

197

TUNING AND TEMPERAMENT

versity of Iowa1" have shown that there is no such thing as sta-
bility of pitch among singers: scooping is found in almost half
the attacks and averages a whole tone in extent; portamento is
very common; the sustained part of the pitch varies from the
true pitch by a comma or more in one-fourth of the notes ana-
lyzed. If we add to these errors the omnipresent vibrato, with
an average extent of a semitone, it would seem that the ambitious
and optimistic director of an unaccompanied choir has an impos-
sible task.

Let us assume, for the moment, that it is possible for a choir
to sing without these pitch fluctuations, that all its members can
sing a note a comma higher or lower when necessary, and that
the director has analyzed the music and marked the places where
the comma shifts are to be made. What have we then? Strangely
enough, if the harmony consists of simple diatonic progressions,
typical of the seventeenth and eighteenth centuries, the pitch will
probably falL With modal progressions, as in Palestrina, it is
more likely to remain stationary. According to GustavEngel, if
one were to consider possible comma shifts whenever a modu-
lation occurs, most of the recitatives in Mozart's Don Giovanni
would fall from one to four commas if sung unaccompanied, and
the final pitch of the opera would be five or six semitones flatter
than at the beginning, A or A*3 instead of D!

If the music contains much chromaticism and remote modu-
lations, even the best-trained choir would probably flounder.
And yet there are choral compositions of the sixteenth and early
seventeenth centuries that seem strikingly modern because of
these very features. De Rore's madrigal "Calami sonum fer-
entes" for four basses (c. 1555) begins with an ascending chro-
matic scale passage treated in imitation. Later it has a re-
markable faburden of inverted major triads a semitone apart — G
F# G AD G. Caimo's madrigal "E ben raggion" (1585) contains
a very smooth example of modulation in which the F* major triad
is heard, and, 24 bars later, its enharmonic equivalent, the G"
major triad. In just intonation the latter triad would be a large
diesis (42 cents, or almost a quarter tone) higher than the former.

16Carl E„ Seashore, The Vibrato (Iowa City, Iowa, 1932).
198

FROM THEORY TO PRACTICE

And what of Marenzio's madrigal "O voiche sospirate a mig-
liornote,'' where there is a modulation around the circle of fifths
from C to G", an enharmonic change from Gb to F#, and further
modulation on the sharp side? According to Kroyer, from whom
all these examples have been taken, this is the first time in mu-
sic that the circle of fifths has been completed. *' Could Mar-
enzio's madrigal have been sung in just intonation?

Gesualdo has the respect of the moderns because of his har-
monic freedom. The best known of his chromatic madrigals is
the "Resta di darmi noia," in which he passes from G minor to
E major, and then sequentially from A minor to F# major. Lis-
ten to the recording of this madrigal by a group of unaccompa-
nied singers in the album 2000 Years of Music and you will prob-
ably agree that the attempt to record it was a noble experiment
and nothing more.

Of course the point that is missed by all these rabid expo-
nents of just intonation in choral music is that this music was
not ordinarily sung unaccompanied in the sixteenth century. A^
cappella meant simply the absence of independent accompani-
ment, not of all accompaniment. If a choir usually sang motets
accompanied by an organ in meantone temperament, it would
quickly adapt itself to the intonation of the organ. If this choir
were in the habit of singing madrigals accompanied by lutes or
viols in equal temperament, its thirds would be as sharp as the
thirds are today. Kroyer thought the pronounced chromaticism
of the Italian madrigalists showed the influence of keyboard in-
struments. On the contrary: it must have been the fretted in-
struments, already in equal temperament, that influenced com-
posers like de Rore, Caimo, Marenzio, and Gesualdo to write
passages in madrigals that could not have been sung in tune with-
out accompaniment.

Present Practice

What is tuning like today? A generation ago, Anglas made
some excellent observations about the intonation of the symphony

i n

1(Theodor Kroyer, "Die Anfange der Chromatik im italienischen Madrigal
des XVI. Jahrhunderts," Internationale Musikgesellschaft, Beiheft 4 (1902).

199

TUNING AND TEMPERAMENT

orchestra. 1° The pedals of the harp are constructed to produce
the semitones of equal temperament; therefore, once the harp is
put in tune with itself, it, and it alone of all the instruments, will
be in equal temperament The violins show a tendency toward
the Pythagorean tuning, both because of the way they are strung
and because of the players' tendency to play sharps higher than
enharmonic flats. Furthermore, in a high register both the vio-
lins and the flutes are likely to play somewhat sharp for the sake
of brilliance. He might have added that the brass instruments,
making use of a more extended portion of the harmonic series
than the woodwinds, have a natural inclination toward just into-
nation in certain keys. The result is "a very great lack of pre-
cision," with heterogeneous sounds that are a mixture of "just,
Pythagorean, tempered, or simply false." Of course the ears of
the audience, trained for years to endure such cacophony, ac-
tually are pleased by what seems to be a good performance.

LI. S. Lloyd has written an article with the frightening title
"The Myth of Equal Temperament."1^ It would be pretty dis-
couraging for the present author to have done extended research
upon the history of equal temperament only to learn at last that
his subject matter was in the class with the story of Cupid and
Psyche! But Lloyd has not actually consigned equal temperament
to the category of the tale of George Washington and the cherry
tree. His argument is against rigidity of intonation, the rigidity
that is inherent in any fixed system of tuning. He holds that the
players in a string quartet or the singers in a madrigal group
are likely to be guided by the music itself as to what intonation
to use, sometimes approaching Pythagorean intervals when me-
lodic considerations are paramount or just intervals when the
harmony demands it. And undoubtedly this freedom of intonation,
plus a well-defined vibrato, does increase the charm of these
more intimate chamber ensembles.

Not even the piano is exempt from the charge of inexactness.
Three-quarters of a century ago Alexander Ellis showed that the

18J. P. L. Anglas, Precis d' acoustique physique, musical, physiologique
(Paris, 1910), p. 206.

19Music and Letters, XXI (1940), 347-361.
200

FROM THEORY TO PRACTICE

best British tuners of his day failed to tune pianos in equal tem-
perament within desirable limits of error. There is no reason
to believe that modern British tuners, or American ones either,
are doing a better job than was done then. Schuck and Young
even show that, because of the inharmonicity of the upper par-
tials of the piano, a tuner is bound to tune the upper octaves pro-
gressively sharper and the lowest octaves progressively flatter
than those in the middle range. *® Their theoretical findings
agree with measurements Railsback had already made of pianos
tuned in equal temperament. However, the psychologists tell us
that "stretched" octaves at top and bottom are a concomitant of
normal hearing. Therefore the sharpness and the flatness re-
spectively would probably be heard as correct intonation.

Now all of this paints a dismal picture. Apparently nobody —
not the pianist, nor the singer, nor the violinist, nor the wind-
player — is able to perform in correct equal temperament. The
harpist is left sitting alone, but no doubt he will be joined by the
Hammond organist, whose instrument comes closest to the equal
tuning.

This contemporary dispute about tuning is perhaps a tempest
in a teapoto It is probably true that all the singers and players
are singing and playing false most of the time. But their errors
are errors from equal temperament. No well-informed person
today would suggest that these errors consistently resemble de-
partures from just intonation or from any other tuning system
described in these pages. Equal temperament does remain the
standard, however imperfect the actual accomplishment may be.

The trend of musical composition during the late nineteenth
and the first half of the twentieth century has been to exploit the
resources of equal temperament, of an octave divided into 12
equal parts, and hence also into 2, 3, 4, or 6 parts. To ascertain
how far back this trend extends is not the purpose of this book.
It would be foolish to deny that this modern trend is different in
kind from the progressions of classic harmony, progressions
that were almost as common in 1600 as in 1800. But it may be
denied that these classic progressions were intimately connected

20O. H. Schuck and R. W. Young, "Observations on the Vibrations of Piano
Strings," Journal of the Acoustical Society of America, XV (1943), 1-11.

201

TUNING AND TEMPERAMENT

with the meantone temperament, as has often been alleged; for
we have seen that the original 1/4 -comma meantone system did
not even reign supreme in 1600, much less in 1700 or 1750. In
1600 there were half a dozen or more ways to tune the octave;
in 1732 Neidhardt gave his readers a choice of twenty! Moreover,
there is every reason to believe that in practice there were far
greater departures from these extremely varied tuning methods
of the seventeenth and eighteenth centuries than there are from
equal temperament today.

In the very nature of things, equal temperament has undergone
vicissitudes during the last four hundred years, and will continue
to do so. Perhaps the philosophical Neidhardt should be allowed
to have the last word on the subject: "Thus equal temperament
carries with itself its comfort and discomfort, like the holy es-
tate of matrimony." 21

^*Gantzlich erschopfte, mathematische Abtheilungen, p. 41.
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TUNING AND TEMPERAMENT

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207

TUNING AND TEMPERAMENT

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209

TUNING AND TEMPERAMENT

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210

LITERATURE CITED

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211

TUNING AND TEMPERAMENT

Malcolm, Alexander. A Treatise of Musick. Edinburgh, 1721.
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Harmonie universelle. Paris, 1636-37.

Metius, Adrian. (There is an inexplicit reference to him in
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Montucla, Jean Etienne. Histoire des mathematiques . New ed.,
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Montvallon, Andre Barrigue de. Nouveau systeme de musique
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Morley, Thomas. A Plaine and Easie Introduction to Practicall
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Nassarre, Pablo. Escuela musica, Part I. Zaragoza, 1724.

Naylor, Edward Woodall. An Elizabethan Virginal Book. Lon-
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Neidhardt, Johann George. Beste und leichteste Temperatur des
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LITERATURE CITED

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Pachelbel, Johann. (His clavier compositions are reprinted in
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213

TUNING AND TEMPERAMENT

Printz, Wolff gang Caspar. Phrynis Mytilenaeus oder der saty-
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"Pro clavichordiis faciendis. "A fifteenth-century Erlangen man-
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The reference in the present study is to Vol. XIV of
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Reinhard, Andreas. Monochordum. Leipzig, 1604.

Riemann, Hugo. Geschichte der Musiktheorie. Berlin, 1898.
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Robet-Maugin, J. C. Manuel du Luthier. Paris, 1834. (Reference
in Garnault's Le temperament.)

Romberg, Bernard. Violoncell Schull. Berlin, 1840 (?)•

Romieu, Jean Baptiste (?). "Memoire theorique & pratique sur
les syst^mes temperas de musique," Me'moires de
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Rossi, Lemme. Sistema musico. Perugia, 1666.

Rousseau, Jean, Traite* de la viole. Paris, 1687.

Rousseau, Jean Jaques. Dictionnaire de musique. Paris, 1768.
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Sabbatini, Galeazzo. Regola facile e breve per sonare sopra il
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Sancta Maria, Tomas de. Arte de taner fantasia. Valladolid,
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214

LITERATURE CITED

Sauveur, Joseph. "Systeme general des intervalles des sons,"

Me'moires de Tacade'mie royale des sciences (1701),

pp. 403-498.
"Table geWrale des systemes tempe're's de musique,"

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Schneegass, Cyriac. Nova & exquisita monochordi dimensio.

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215

TUNING AND TEMPERAMENT

Spitta, Philipp. Johann Sebastian Bach. 2 vols. Translated by
Clara Bell and J. A. Fuller-Maitland. London, 1884.
Vol. I.

Stanhope, Charles, Earl. "Principles of the Science of Tuning In-
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are in Denkmaler deutscher Tonkunst, IV Band, 1.
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Vierteljahrsschrift fur Musikwissenschaft, VI (1890),
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des Herrn Kirnberger. Berlin and Leipzig, 1775.

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Torchi, Luigi. L'arte musicale in Italia, Vol. III. Milan, post
1897.

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Verheijen, Abraham. See Stevin' s Van de Spiegeling der Sing-
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216

LITERATURE CITED

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Warren, Ambrose. The Tonometer. London, 1725.

Wasielewski, Joseph Wilhelm von. Geschichte der Instrumental -
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Werckmeister, Andreas. Hypomnemata musica. Quedlinburg,
1697.

Musicalische Paradoxal-Discourse. Quedlinburg,

1707.

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1691.

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Wiese, Christian Ludwig Gustav, Baron von. Klangeintheilungs- ,
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Young, Thomas. "Outlines of Experiments and Inquiries Re-
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Zacconi, LodovicOo Prattica di musica, Part I. Venice, 1592.

Zarlino, Gioseffo. Dimostrationi armoniche. Venice, 1571.

Istitutioni armoniche. Venice, 1558.

Sopplimenti musicali. Venice, 1588.

Zeising, Heinrich. Theatri machinarum. Altenburg, 1614.

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XXXVI. Milan, 1919.

217

INDEX

abacus Triharmonicus, 109 f.

Abrege des regies de l'harmonie. See Levens.

Adlung, J., 85.

Agricola, M., 4, 10, 95, 149-151.

Akustik. See C. E. Schumann.

Alexander the Great, 122.

Allgemeine Theorie der Musik. See F. W.

Opelt.
Allgemeine Theorie der Schonen Kiinste. See

J. G. Sulzer.
Amiot, J. M., 77.
"Die Anfange der Chromatik im italienischen

Madrigal des XVI. Jahrhunderts." See T.

Kroyer.
Anglas, J. P. L., 197 f.
Anleitung zu den musikalischen Gelahrtheit.

See J. Adlung.
Anleitung zur Tonometrie. See J. D. Berlin.
Anonymous, author of Exposition de quelques

nouvelles vues mathematiques, 125.
Anonymous, author of Pro clavichordiis fa-

ciendis, 91 f.
L'antica musica ridotta alia modernaprattica.

See N. Vicentino.
Anweisung wie man Claviere ... stimmen

konne . See B. Fritz,
approximations to equal temperament. See

temperament, equal: geometrical and

mechanical approximations, and numerical

approximations .
approximations to the meantone temperament.

See temperament, meantone: approxima-
tions.
Appun, G., 119.
Arabian scale. See multiple division: equal

divisions: 17-division.
Archicembalo, 27, 115 f, 152.
Archicymbalam, 106.
Archimedes, 34, 50.
Archytas, 16 f, 19, 22 f, 143.
Ariel, 113.

Aristoxenus, 2, 16 f, 19, 22-24, 57.
arithmetical division. See division, arithmet-
ical.
L'arithmetique des musiciens. See J. E.

Gallimard.
Aron, P., 10, 26, 49.
L'art du facteur d'orgues. See F. Bedos.
Arte de taner fantasia. See T. de Sancta

Maria.
L'arte musicale in Italia. See L. Torchi.
Artusi, G. M., 8, 10, 46,_IT2, 144-148, 176,

186.
Awraamoff, A. M., 24, 152.

"Bach and The Art of Temperament." See J.

M. Barbour.
Bach, C. P. E., 47 f.

Bach, J. N., 85-87.

Bach, J. S., 10, 12 f, 85-87, 146, 189, 192-194.

Bakfark, V., 185.

Ballet comique de la reine, 8.

Barbour, J. M., 3, 77, 112, 131, 161, 189-197.

Barca, A., 42 f.

Bartolus, A., 142 f.

Beaugrand, J. de, 79, 81, 84.

Bedos, F-, 125.

Beer, J., 124.

bells, 7, 142.

Bendeler, J. P., 157-160.

Berlin, J. D., 119.

Bermudo, J., 3, 5, 46, 137, 162-164, 186.

Bertelsen, N. P. J., 127.

Beste und leichteste Temperatur des Mono-

chordi. See J. G. Neidhardt.
Blankenburg, Q. van, 105 f, 118.
Bohme, F. M., 187.
Boethius, A. M. S., 3, 121.
bonded clavichord. See clavichord, bonded.
Bonnet, J., 186.
Bosanquet, R. H. M., 9, 32, 114, 117, 119-121,

123, 125-127, 129 f, 131.
Bossier, H. P., 49 f.
Bottrigari, E., 8, 46, 147 f.
Boulliau, I., 54, 79-81.
Bumler, G. H., 80.
Bull, J., 187 f, 192 f.
Buttstett, J. H., 105.
Byrd, W., 188.

Cahill, T., 74.

Caimo, J., 196 f.

Caramuel, J., 3.

Cardano, G., 57.

Caus, S. de, 11, 95 f, 101.

cembalo. See keyboard instruments.

cent, ii and passim.

Cerone, P., 46.

Cherubini, M. L-, 58.

China, 7, 55 f, 77-79, 122, 150 f.

"Chine et Coree." See M. Courant.

Choquel, H. L., 152.

choral music. See just intonation in choral
music.

chromatic genius. See Greek tunings.

chromaticism, 187-189, 196 f.

circle of fifths, 106, 188-194, 197.

circulating temperaments. See irregular sys-
tems: circulating temperaments.

clavichord. See keyboard instruments.

, bonded, 30 f, 145-147.

clavier. See keyboard instruments.

closed system. See temperament, regular.

Cogitata physico-mathematica. See M. Mer-
senne.

Colonna, F., 23 f, 151-154.

219

TUNING AND TEMPERAMENT

column of differences. See tabular differ-
ences.

comma, i and passim. See especially irregu-
lar systems: divisions of ditonic comma.

Compendio de musica. See D. Varella.

Compendio del trattato de' generi, e de' modi-
See G. B. Doni.

A Compleat Method

See G. Keller.

The Compleat Tutor for the Harpsichord or
Spinet. See P. Prelleur.

continued fractions, 54, 74, 124, 128-130.

correspondences between equal multiple divi-
sions and varieties of meantone tempera-
ment, 124.

Couperin, F., 191 f.

Courant, M., 55 f, 122, 150.

Critica musica. See J. Mattheson.

Crotch, W., 31 f.

cube root. See duplication of the cube; loga-
rithms; mesolabium.

Cursus seu mundus mathematicus. See R. P.
C. F. M. Dechales.

De institutione musica. See A. M. S. Boethius.
De la musique des Chinois. See J. M. Amiot.
De musica libri VII. See F. Salinas,
decameride, 120, 134.
Dechales, R. P. C. F. M., 25, 36.
Declaracion de instrumentos musicales. See

J. Bermudo.
Delannoy, 58.
Delia imperfettioni della moderna musica.

See G. M. Artusi.
Delezenne, C. E. J., 58, 120.
Denis, J., 47.
Descartes, R., 53.
U desiderio. See E. Bottrigari.
deviation, iii and passim.
Dialogo della musica antica e moderna. See

V. Galilei,
diaschismata, 121.
diatonic genus. See Greek tunings.
Dictionnaire de musique. See J. J. Rousseau.
Didymus, 2, 18, 20 f, 23, 887109, 152, 195.
diesis, 10, 108, 196.

Dimostrationi armoniche. See G. Zarlino.
Discorsi ... e due nuove scienze. See G. Gal-
ilei,
ditone, 21 f, 115.
ditonic comma, i and passim. See especially

irregular systems: divisions of ditonic

comma,
division, arithmetical, 21, 29 f, 56 f, 60-64,

68-73, 80, 83, 85, 99, 139-144, 150, 155,

162, 175.
division, geometrical, 38 f, 80, 92 f, 150, 154.

See also Euclidean construction.
Don Giovanni. See W. A. Mozart.

Doni, G. B., 109 f.

Douwes, C, 30, 146.

Dowland, J., 4, 24, 151, 153.

Dowland, R., 151.

Drobisch, M. W., 37, 114, 120, 123, 125, 127-

129.
dulcimer, 112, 190.
duplication of the cube, 49.
Dupont, W., 91, 105.
Durham Cathedral, 106.

"Eine mathematisch-harmonische Analyse des

Don Giovanni von Mozart." See G. Engel.
Eitz, K. A. ii.

Elementa musica. See Q. van Blankenburg.
Elementa musicalis. See Faber Stapulensis.
Elementarbuch der Tonkunst. See H. P.

Bossier.
An Elementary Treatise on Musical Intervals

and Temperament. See R. H. M. Bosanquet.
Elements of Musical Composition. See W.

Crotch.
An Elizabethan Virginal Book. See E. W.

Naylor.
Ellis, A. J., ii, 48 f, 65, 73-76, 87, 111, 131,

146, 193-195, 198 f.
Elsasz, 113.
Engel, G., 196.

enharmonic genus. See Greek tunings.
Enharmonic Organ, 110 f.
Enharmonium, 111.
eptameride, 120.

equal temperament. See temperament, equal.
Eratosthenes, 16, 18 f, 22 f.
Erlangen University Library, 91.
Escuela musica. See P. Nassarre.
espinette, 47.
An Essay upon Perfect Intonation. See H. Lis-

ton.
Euclidean construction, 27, 50, 52-55, 93, 137,

142, 144.
Euharmonic Organ, 110.
Euler, L., 65, 100 f, 109 f.
Eutocius, 50.
exponents, ii, 95, 102.
Exposition de quelques nouvelles vues mathe-

matiques dans la theorie de la musique,

125.
Exposition d'une nouvelle methode pour l'en-

seignement de la musique. See P. Galin.
extensions of just intonation. See multiple di-
vision: extensions of just intonation.
Extract aus der neuen Musica oder Singkunst.

See D. Hizler.

Faber Stapulensis (Jacques le Febvre), 137,

144.
Faggott, J. 65-67.

220

INDEX

Farey, J., 65, 134.

Farnaby, G., 188.

Faulhaber, J., 78, 80.

Febvre, J. le. See Faber Stapulensis.

Fetis, F. J., 114.

Fibonacci series, 115.

Fischer, J. P. A., 7.

Fitzwilliam Virginal Book, 188.

Fludd, R., Frontispiece, 3.

flute. See wind instruments.

Fogliano, L., 11, 92-96, 104, 106.

Foundling Hospital, 106.

"Frequency Ratios of the Tempered Scale."
See C. Williamson.

Frescobaldi, G., 189, 192.

fretted clavichord. See clavichord, bonded.

fretted instruments, 6-8, 11, 25, 28, 40, 42,
45 f, 50, 57-59, 98 f, 103, 139-142, 144-149,
151, 162-164, 182-186, 188, 197.

fretted instruments, in sixteenth century paint-
ings, 12.

Fritz, B., 47 f.

Froberger, J. J., 190.

Fronimo. See V. Galilei.

Fuller-Maitland, J. A., 188.

Gafurius, F., 3, 5, 25.

Galilei, G., 11.

Galilei, V., 8, 46, 57-64, 149, 184.

Galin, P., 31.

Galle (J. B. Gallet), 79, 81, 84.

Gallimard, J. E., 12, 118, 134 f, 137 f.

Ganassi, S., 10, 24, 68, 70, 139-143, 154, 176,
186.

Gantzlich erschopfte, mathematische Abthei-
lungen des diatonisch-chromatischen, tem-
perirten Canonis Monochordi. See 3. G.
Neidhardt.

Garnault, P., 58 f.

Gedanken iiber die Temperatur des Herrn
Kirnberger. See G. F. Tempelhof.

generalized keyboard, 9, 117, 130.

G^ne'ration harmonique. See J. P. Rameau.

geometrical approximations. See tempera-
ment, equal: geometrical and mechanical
approximations.

geometrical division. See division, geometri-
cal.

Geschichte der Instrumentalmusik im 16.
Jahrhundert. See J. W. Wasielewski.

Geschichte der musicalischen Temperatur.
See W. Dupont.

Geschichte der Musiktheorie. See H. Riemann.

Gesprach zwischen einem Musico theoretico
und einem Studioso musices. See G. A.
Sorge.

Gesualdo, C., 197.

Gibbons, O., 187, 189, 192.

Gibelius, O., 29, 85.

Gintzler, S., 185.

Glyn, M. H., 186.

Goetschius, P., 4.

golden system, 127.

Das goldene Tonsystem ... . See T. Kornerup.

Gonzaga, 117.

"good" temperaments. See irregular systems:

circulating temperaments.
Gow, J., 50 f.
Grammateus, H., 3, 6, 10, 13, 25, 46, 69, 137-

139, 142, 144, 151, 157, 176, 186.
Greek tunings, 15-24.

diatonic, 15, 19-21.

chromatic, 15, 17 f, 21.

enharmonic, 15 f, 21, 109, 115.

enharmonic, modern, 15,33 f, 188, 190, 193,
198.

enharmonic of Salinas, 106 f.
Greene, P. C., 195.
Grondig Ondersoek van de Toonen der Musijk.

See C. Douwes.
Guido of Arezzo, 25, 36.
guitar. See fretted instruments.

hackebort. See dulcimer.

Haser, A. fTTTOI.

Hammond, L., 74-76.

Hammond Electric Organ, 74-76, 199.

Handel, G. F., 10, 106.

Harmonices mundi. See J. Kepler.

Harmonicorum libri xn. See M. Mersenne.

Harmonicorum libri tres. See C. Ptolemy.

Harmonics. See R. Smith.

Harmonie universelle. See M. Mersenne.

harmonium. See keyboard instruments.

harp, 198 f.

harpsichord. See keyboard instruments.

Harrison, 40 f.

Hawkes, W., 133 f, 137 f.

Helmholtz, H. L. F., 48, 73 f , 89, 109 f.

Henfling, K., 120.

Hero (Heron) of Alexandria, 53.

Herschel, J., 129.

Hindoo scale. See multiple division: equal

divisions: 22-division.
Histoire des mathe'matiques. See J. E. Mon-

tucla.
Histoire generate de la musique. See F. J.

Fetis.
Hizler, D., 118.
H& Tchheng-thyen, 55 f, 150.
Holbein, H., 12.
Holder, W., 47, 123.
Hugo de Reutlingen, 88.
Hutton, C., 50.

Huyghens, C., 9, 37, 116-118.
Hypomnemata musica. See A. Werckmeister.

221

TUNING AND TEMPERAMENT

Ingenieurschul. See J. Faulhaber.

inharmonic ity of upper partials, 199.

instruments. See fretted instruments, key-
board instruments, stringed instruments,
wind instruments; also, bells, dulcimer,
harp.

"Intonation of Solo and Ensemble Perfor-
mance." See J. F. Nickerson.

"Introduzione a una nuova teoria di musica."
See A. Barca.

irregular systems, 4, 6, 10, 24, 32, 131-182,
200.

irregular systems: circulating temperaments:
12 f , 155, 165, 168, 176-182, 192 f.

irregular systems: circulating temperaments:
Out-of-Tune Piano, 161.

irregular systems: circulating temperaments:
temperament by regularly varied fifths, iii,
179-182.

irregular systems: divisions of ditonic com-
ma, 12, 44, 84, 154-175. 1/2 comma, 138,
155-157; 1/3, 157-160; ~tj\, 159 f; 1/5,
159-161; 1767 161-164, 170, 179; 1/6, l/i,
171-173; 177, 164; 1/12, 1/4, 171 f; 1/12,
1/6, 136, 165-172, 178 f; 1/12, 1/6, 1/4",
173-175; 1/12, 3/16, 166; 1/12, 5/24, 170 f.

irregular system: Metius' system, 175 f.

irregular systems: modifications of regular
temperaments, 24, 137-149.

irregular systems: modifications of regular
temperaments: Pythagorean, 46, 137 f,
186.

irregular systems: modifications of regular
temperaments: just, 139-144, 186.

irregular systems: modifications of regular
temperaments: meantone, 12, 46, 131-135,
144-149, 179, 186.

irregular systems: temperaments largely
Pythagorean, 12, 61, 149-154.

"Irregular Systems of Temperament." See
J. M. Barbour.

Istitutioni armonische. See G. Zarlino.

Jackson, W., 125.

Jacobi, K. G. J., 129.

Jank6, P. von, 120.

Jeans, J., 4.

"Jenseits von Temperierung und Tonalitat."
See A. M. Awraamoff.

Johann Sebastian Bach. See P. Spitta.

Judenkunig, H., 185 f.

just intonation, 2, 4, 9-11, 21, 88-104, 121,
131, 176, 199.

just intonation, extensions of. See multiple
division: extensions of just intonation.

just intonation in choral music, 194-197.

just intonation, modifications of. See irregu-
lar systems: modifications of regular
temperaments.

just intonation, theory of, 101-104.

"Just Intonation Confuted." See J. M. Barbour.

Keller, G., 47.

Kepler, J., 11, 58-60, 96 f, 99.

keyboard instruments, 6-9, 25-48, 61, 89,
97-99, 103, 105-123, 130, 135 f, 139, 142,
147-149,186-195,197-199. See also abacus
Triharmonicus, Archicembalo, Archicym-
balam, Enharmonic Organ, Enharmonium,
Euharmonic Organ, Hammond Electric Or-
gan, Pantalonisches Cimbal, Telharmoni-
um.

King FSng; 122.

Kinkeldey, O., iv, 26, 45, 186.

Kircher, A., 9, 52 f, 108, 115, 122-124.

Kirnberger, J. P., 12, 64 f, 155-157, 163.

Kbrte, O., 183.

Klangeintheilungs-, Stimmungs- und Temper-
atur-Lehre. See C. L. G. von Wiese.

Kornerup, T., 113, 121, 127 f, 195.

Kroyer, T., 197.

Kuhnau, J., 190 f.

Die Kunst des reinen Satzes in der Musik.
See J. P. Kirnberger.

Kurze Anflihrung zum General- Bass. See J.
H. Buttstett.

La Laurencie, L. de, 71.

Lambert, J. H., 65.

Lanfranco, G. M., 11, 45 f, 186.

"Laute und Lautenmusik bis zur Mitte des 16.

Jahrhunderts." See O. Korte.
Levens, 143-145.
least squares, 127 f.

linear correction. See division, arithmetical.
Liston, H., 110 f.
Lloyd, LI. S., 198.
Lobkowitz. See J. Caramuel.
logarithms, 3, 9, 30, 41, 64, 72 f, 77-79, 116,

118, 120, 128 f, 134, 154, 176.
lute. See fretted instruments.

madrigal, 144, 196 f.

Malcolm, A., 3, 99 f, 141, 143.

Manuel du luthier. See J. C. Robet-Maugin.

Marchettus of Padua, 118.

Marenzio, L., 144, 197.

Marinati, A., 45 f.

Marpurg, F. W., ii, 12, 43 f, 53, 65, 68, 84,
88, 98-100, 103 f, 139, 143, 154-182 (pas-
sim).

Mathematical Dictionary. See C. Hutton.

Mathesis nova. See J. Caramuel.

Mattheson, J., 124, 192.

Marziale, M., 12.

mean proportional. See Euclidean construc-
tion.

222

INDEX

mean-semitone temperament. See irregular
systems: modifications of regular temper-
aments.

meantone temperament. See temperament,
meantone.

mechanical approximations. See tempera-
ment, equal: geometrical and mechanical
approximations.

Meckenheuser, J. G., 80, 83.

Melder, J., 78.

"Memoire sur les valeurs numeriques des
notes de la gamme." See C. E. J. Dele-
zenne.

"Memoire th£orique & pratique sur les sys-
temes tempe're's de musique." See J. B.
Romieu.

Mercadier, J. B., 135, 166 f.

Mercator, N., 9, 118, 123.

Merian, W., 187.

meride, 120.

Mersenne, M., iii, 7, 9, 11 f, 48, 51-55, 58 f,
61, 74, 79 f, 84, 87, 97-99, 103, 106-108,
118 f, 122, 132-134, 137, 183, 190.

Merula, T., 187, 189.

mesolabium, 6, 33, 50 f, 59, 118 f, 144.

Metius, A., 175 f, 181.

Micrologus. See A. Ornithoparchus.

Miller, D. C.,~85.

Molth^e, 52.

monochord, passim.

Monochordum. See A. Reinhard.

monopipe, 85-87.

Monteverdi, C., 8.

Montucla, J. E-, 89.

Montvallon, A. B. de, 100 f .

Morley, T., 151.

Mozart, W. A., 196.

multiple division, 105-130. See also split
keys.

multiple division: equal divisions, 111-130;
17-division, 112, 120, 123, 126, 128; 19-,
9, 34, 112-114, 120 f, 127, 129 f; 22-, 114 f,
119, 128-130; 24-, 115; 25-, 129; 28-, 129;
29-, 115, 120, 125 f, 128; 3L-, 9, 31, 36 f,
42, 51, 115-121, 127-129, 152; 34-, 37, 119,
129; 36-, 119; 41-, 119 f, 126, 128 f; 43-,
36, 118, 120, 123, 126, 128; 46-, 128; 50-,
33, 41 f, 120 f, 127; 51-, 128; 53-, 9, Tl8,
120-123, 125 f, 128-130; 55-, 42 f, 120,
122-126, 191; 56-, 119, 125 f; 58-, 125;
. 65-, 125 f, 128; 67-, 122, 125; 70-, 126,
128; 74-, 37, 119, 125, 128; 77-, 126, 128;
79-, 125; 81-, 127; 84-, 125; 87^, 119, 125 f,
129; 89-, 126, 128; 9L-, 125; 94-, 126, 128;
98-, 125; 105-, 125; 112-, 125; 117-, 125;
118-, 125-129; 131-, 127; 142-, 128; 166-,
128; 212-, 127; 306-, 128; 343-, 127; 559-,
129; 612-, 129; 665-, 128 f; 817-, 129.

multiple division: equal divisions: corres-

pondences with varieties of meantone tem-
perament, 124.

multiple division: extensions of just intona-
tion, 79, 106-112.

multiple division, theory of, 126-130.

"Music and Ternary Continued Fractions."
See J. M. Barbour.

Musica instrumentalis deudsch. See M. Agri-
cola.

Musica mathematica. See A. Bartolus.

Musica mechanica organoedi. See J. Adlung.

Musica practica
Musica theorica

See B. Ramis.
See L. Fogliano.

Musicae activae Micrologus. See A. Ornitho-
parchus.

"Musical Logarithma." See J. M. Barbour.

Musicalische Paradoxal-Discourse. See A.
Werckmeister.

Musicalische Temperatur. See A. Werck-
meister.

La musique rendue sensible par la mechan-
ique. See H. L. Choquel.

Musurgia universalis. See A. Kircher.

"The Myth of Equal Temperament." See LI.
S. Lloyd.

Nassarre, P., 61-64.

National Gallery in London, 12.

"Die natiirliche Stimmung in der modernen

Vokalmusik." See M. Planck.
Naylor, E. W., 188-
negative system, 112-125 (passim).
Neidhardt, J. G., ii, 12, 44, 78, 80, 82, 85-87,

115, 118, 154-182 (passim), 192 f, 200.
New Cyclopedia. See A. Rees.
Ayn new kunstlich Buech. See H. Grammateus.
"A New Theory of Untempered Music." See

N. L. Norden.
Newton, I., 53.
Nickerson, J. F., 195.
Nicomedes, 53.
Nierop, D. R. van, 175.
"Nierop's Hackebort." See dulcimer.
Norden, N. L., 195.
Nouveau systeme de musique ... . See A. B.

Montvallon.
Nouveau systeme de musique theorique. See

J. P. Rameau.
Nouveau systeme de musique theorique et

pratique. See J. B. Mercadier.
Nova & exquisita Monochordi Dimensio. See

C. Schneegass.
"Novus cyclus harmonicus." See C.Huyghens.
numerical approximations. See temperament,

equal: numerical approximations; also,

temperament, meantone: approximations.
"Nytt Pifund, til at finna Temperaturen, i

stamningen for thonerne pa Claveretock

dylika Instrumenter." See D. P. Strahle.

223

TUNING AND TEMPERAMENT

"Observations on the Vibrations of Piano
Strings." See O. H. Schuck andR. W. Young.

Odington, W., 3.

Oettingen, A. von, 119.

omega (o), 127.

"On a New Mode of Equally Tempering the
Musical Scale." See J. Farey.

"On music." See J. Farey.

"On Perfect Harmony in Music ... ." See H.
W. Poole.

"On Perfect Musical Intonation." See H. W.
Poole.

"On the History of Musical Pitch." See A. J.
Ellis.

"On the Musical Scales of Various Nations."
See A. J. Ellis.

On the Principles and Practice of Just Intona-
tion. See P. Thompson.

Opelt, F. W., 99, 114, 119 f.

Orfeo. See C. Monteverdi.

See keyboard instruments.
See J. P. Bendeler.

organ.

Organopoeia.

Orgelund Klavier in der Musik des 16.

Jahr-

hunderts. See O. Kinkeldey.

Ornithoparchus, A., 3, 151.

"Other Necessary Observations to Lute-Play-
ing." See J. Dow land.

"Outlines of Experiments and Inquiries Re-
specting Sound and Light." See T. Young.

Out-of-Tune Piano. See irregular systems:
circulating temperaments: Out-of-Tune
Piano.

Oystermayre, J., 188.

Pachelbel, J., 190, 193.

paintings of the sixteenth century, 12.

Palestrina, G. P. da, 196.

Pantalonisches Cimbal, 190.

Papius, A., 3.

Pappius of Alexandria, 50.

parfait diapason of Mersenne, 106-108.

Parry, H., 192.

Parthenia, 187.

"The Persistence of the Pythagorean Tuning

System." See J. M. Barbour.
Pesarese, D., 33, 113.
Philander, W., 118.
Philo of Byzantium, 51.
Philolaus, 121.
The Philosophy of Musical Sounds. See R.

Smith.
Phrynis Mytilenaeus ... . See W.C. Printz.
pi (it), 40 f, 77.

piano. See keyboard instruments.
Piano Tuning and Allied Arts. See W. B.

White.
A Plaine and Easie Introduction to Practicall

Musicke. See T. Morley.

Planck, M., 11.

Plato, 53.

Poole, H. W., 110.

Populare Darstellung der Akustik. See H.
Riemann.

positive system, 112-125 (passim).

Practica musica. See F. Gafurius.

Praetorius, M., 9, 28 f, 62, 113.

Prattica di musica. See L. Zacconi.

Precis d'acoustique .... See J. P. L. Anglas.

Predis, A. de, 12.

Prelleur, P., 47.

present practice of tuning, 197-200.

"Present-Day Preferences for Certain Mel-
odic Intervals." See A. M. Small.

"Principles of the Science of Tuning Instru-
ments with Fixed Tones." See C. Stanhope.

Printz, W. C, 29, 37, 119, 142 f.

Pro clavichordiis faciendis, 91 f.

Propositiones mathematico-musicae. See O.
Gibelius.

Prout, E., 4.

Ptolemy, C., 2, 16-23, 57, 88, 152, 195.

Pythagoras, 1, 139.

Pythagorean tuning, 1-4, 10, 21-23, 42, 45, 56,
59, 68, 88-91, 95, 101 f, 110-112, 121 f, 131,
147, 150 f, 176, 183, 194, 198.

Pythagorean tuning, modifications of. See ir-
regular systems: modifications of regular
temperaments, and temperaments largely
Pythagorean.

Railsback, O. L., 199.

Les raisons des forces mouvantes avec di-
ver ses machines. See S. de Caus.

Ramarin, 124.

Rameau, J. P., 4, 11 f, 133, 137, 192.

Ramis, B., 4, 10, 25, 88-92, 104 f, 151, 194.

Redford, J., 186 f.

Rees, A., 65.

Regola facile e breve per sonare sopra il
basso continuo. See G. Sabbatini.

Regola Rubertina. See S. Ganassi.

regular temperament. See temperament, reg-
ular.

Reinhard, A., 68, 141-143, 186.

Das Relativitatsprincip der muslkalischen
Harmonie. See Ariel.

"Remarques sur les temperaments en mu-
sique." See J. H. Lambert.

Riemann, H., 25, 114, 116 f, 119 f, 135.

Roberti, E. de, 12.

Roberval, 52.

Robet-Maugin, J. C, 58.

Romberg, B., 58 f.

Romieu, J. B., 37, 40, 42 f, 101, 114, 119 f,
123-126.

Rore, C. da, 196 f.

224

INDEX

Rossi, L., 29 f, 35 f, 51, 53, 115, 118-120.

Rossi, M., 187 f.

Rousseau, J., 105.

Rousseau, J. J., 99 f, 104.

Roussier, P. J., 4.

Rudimenta musices. See M. Agricola.

Ruscelli, G., 6.

Sabbatini, G., 108.

Sachs, C., iii, 197.

St. Martin's Church in Lucca, 105 f.
Salinas, F., 6, 9, 33-35, 42, 46, 50 f, 106 f,
113, 118, 186.

Salmon, T., 143.

Sambuca Lincea, 151-154.

Sancta Maria, Tomas de, 28.

Per satyrische Componist. See W.C. Printz.

Sauveur, J., 112, 114, 118, 120, 123-126, 129.

scale, Arabian. See multiple division: equal
divisions: 17-.

scale, Hindoo. See multiple division: equal
divisions: 22-.

scale, Siamese, 112.

Scarlatti, D., 191-193.

Scheidt, S., 189.

A Scheme Demonstrating the Perfection and
Harmony of Sounds. See W. Jackson.

Schering, A., 186.

schisma, 64, 80, 89, 92, 110 f, 154, 156.

Schlick, A., 6, 10, 26, 46, 131, 135-139, 168,
181 f, 186 f.

Schneegass, C., 37-40, 119.

Schbnberg, A., 114, 183.

Schola phonologica. See J. Beer.

Schreyber, H. See Grammateus.

Schroter, C. G., 68-73, 77.

Schuck, O. H., 199.

Schumann, K. E., 159.

Science and Music. See J. Jeans.

Scintille de musica. See G. M. Lanfranco.

Scriabin, A., 113.

Seashore, C. E., 196.

Seconda parte dell' Artusi. See G. M. Artusi.

Sectio Canonis harmonici. See J. G. Neidhardt.

semi-meantone temperament. See irregular
systems: modifications of regular tem-
peraments: mean-semitone temperament.

"Ein Sendschreiben liber Temperatur-Berech-
nung." See C. G. Schroter.

Senfl, L., 1857

Sensations of Tone. See H. L. F. Helmholtz;
also, A. J. Ellis.

Septenarium temperament, 164.

Serini, G., 191.

sesqui-. See superparticular ratio.

Seu-ma Pyeou, 122.

sexagesimal notation, 16, 79-81.

A Short History of Greek Mathematics. See
J. Gow.

Siamese scale, 112.

Silbermann, G., 9, 13, 42, 112, 124, 191, 194.

Sistema musico. See L. Rossi.

"A Sixteenth Century Approximation fortr."

See J. M. Barbour.
Small; A. M., iv, 195.
Smith, R., 40-42.

Societats-Frucht. See J. G. Meckenheuser.
Die sogenannte allerneueste musicalisches

Temperatur. See J. G. Meckenheuser.
Somma de tutte le scienza. See A. Marinati.
Sophiae cum moria certamen. See R. Fludd.
Sopplementi musicali. See G. Zarlino.
Sorge, G. A., 42, 83 f, 1247 159, 194.
Spataro, G., 4.
"Specimen de novo suo systemate musico."

See K. Henfling.
Spiegel der Orgelmacher und Organisten. See

A. Schlick.
spinet. See keyboard instruments.
Spitta, P., 85.

split keys, 33-35, 42, 97, 105 f.
square root. See Euclidean construction.
Squire, W. B.,~188.
Stanhope, C, Earl, 157 f, 163.
Steiner, J., 111.
Stella, S., 117.

Stevin, S., 7, 11, 28, 76 f, 79, 192.
Str'ahle, D. P., 65-68.
stretched octaves, 199.
stringed instruments, 4, 8, 45 f, 58 f, 124, 195,

198 f.
"Studien im Gebiete der reinen Stimmung."

See S. Tanaka.
Sulzer, J. G., 65.
superparticular division, 2. See also Intervals

with Superparticular Ratios, the table fol-
lowing this index,
superparticular division, of the tetrachord,

23 f.
superparticular division, of the tone, 154.
Suremain-Missery, A., 166.
Sweelinck, J. P., 189.
symmetry, 155-182 (passim).
Syntagma musicum. See M. Praetorius.
syntonic comma, i and passim.
"Systeme general des intervalles des sons."

See J. Sauveur.

"Table general des syst^mes tempeVe's de

musique. See J. Sauveur.
tabular differences, 68-73.
Tagore, S. M., 114.
Tallis, T., 186 f.

Tanaka, S., 6, 33, 109, 111, 117, 122 f, 135 f.
Der Tanz in den deutschen Tabulaturblichern.

See W. Merian.
Telemann, G. P., 124, 191 f.
Telharmonium, 74.

225

TUNING AND TEMPERAMENT

temperament, 5. See also tunings.

temperament, by regularly varied fifths. See
irregular systems: temperament by regu-
larly varied fifths.

temperament, circulating. See irregular sys-
tems: circulating temperaments.

temperament, equal, 6-8, 10 f, 25, 29, 45-87,
90, 131, 142, 146 f, 164, 176-178, 183-186,
188, 191-195, 197-200.

temperament, equal: geometrical and mechan-
ical approximations, 49-55.

temperament, equal: numerical approxima-
tions, 55-87.

temperament, history of, 1-14.

temperament, meantone, 9-11, 25-44, 71, 106,
115, 124, 142, 176, 189-192, 197, 200.

temperament, meantone: approximations,
29-31, 43.

temperament, meantone: modifications. See
irregular systems: modifications of regu-
lar temperaments.

temperament, meantone, varieties of, 31-44,
131; 1/3 comma, 9, 33-35, 51, 113; 1/5, iii,
35 f, 47, 120, 134; 1/6, 42 f, 112, 124, 135,
146, 168, 176, 191; 1/7, 43; 1/8, 43; 1/9,
44; 1/10, 44; 2/7, 9, 32 f, 35, 37, 46, 50,
119, 121; 2/9, 36 f, 119; 3/10, 40 f; 5/18,
41.

temperament, meantone, varieties of: corres-
pondences with equal multiple divisions,
124.

temperament, "paper," 149, 154, 176.

temperament, regular, 32t44, 91, 112-131.

Le temperament. See P. Garnault.

temperament anacritique, 124.

"Temperament; or, the Division of the Octave."
See R. H. M. Bosanquet.

Tempelhof, G. F., 155.

Temple Church in London, 106.

Tentamen novae theoriae musicae. See L.
Euler.

Theatri machinarum. See H. Zeisung.

Th^orie acoustico-musicale. See A. Sure-
main-Missery.

A Theory of Evolving Tonality. See J. Yasser.

Thompson, P., 110.

Tisdall, W., 188.

Tomkins, T., 188.

The Tonometer. See A. Warren.

Das Tonsystem des Italieners Zarlino. See
T. Kornerup.

Torchi, L., 186.

Toscanello in musica. See P. Aron.

Trait6de l'accord de l'espinette. See J. Denis.

Traits de l'harmonie. See J. P. Rameau.

Traite de la viole. See J. Rousseau.

Transponir-Harmonium, 111.

Treatise ... of Harmony. See W. Holder.

A Treatise of Musick. See A. Malcolm.

trigonometry, 65-67.

trombone. See wind instruments.

Tsai-yii, Prince Chu, 7, 77-79.

tuning, history of, 1-14.

tuning pipe, 85-87.

tuning today. See present practice of tuning.

tunings. See Greek tunings, just intonation,

Pythagorean tuning, etc.
Two Thousand Years of Music. See C. Sachs.

"liber mehr als zwolfstufigegleichschwebende
Temperaturen." See P. von Jank6.

"Uber musikalisches Tonbestimmung und
Temperatur." See M. W. Drobisch.

"liber wissenschafliche Begrundung derMusik
durch Akustik." See A. F. Haser.

unequal temperament. See temperament,
meantone, and temperament, meantone,
varieties of. See also irregular systems.

Van de Spiegeling der Singconst. See S. Stevin.

Varella, D., 58.

Variety of Lute- Lessons. See R. Dowland.

varieties of meantone temperament. See tem-
perament, meantone, varieties of.

Verhandlung van de Klokken en het Klokke-
Spel. See J. P. A. Fischer.

Verheijen, A., 28, 35, 120.

Versuch liber die musikalische Temperatur.
See F. W. Marpurg.

The Vibrato. See C. E. Seashore.

Vicentino, N., 8, 11, 25, 27, 37, 51, 115-119,
142, 152, 183.

vihuela. See fretted instruments.

viol. See fretted instruments.

violin. See stringed instruments.

"Violin Intonation." See P. C. Greene.

Le violon de Lully~a~Viotti. See L. de La
Laurencie.

Violoncell Schull. See B. Romberg.

violoncello. See stringed instruments.

virginals. See keyboard instruments.

Vitruvius, M., 50 f.

voices, 124, 144, 194-197.

Wang Pho, 150 f.

Warren, A., 118.

Wasieleswki, J. W. von, 186.

Wedell, P. S., 127.

Werckmeister, A., 12 f, 105, 154-182 (passim)

192 f.
White, W. B., 48 f.
Wiese, C. L. G., Baron von, 156 f.
Williamson, C., 74-76.
wind instruments, 7, 124, 195, 198.
Wis-konstige Musyka. See D. R. van Nierop.
wolf, 10 f, 27, 34, 91, 131 f, 134, 163.

226

INDEX

Yasser, J., 9, 113 f.

Young, R. W., 199.

Young, T., 12 f, 135, 161, 165 f, 178 f, 181 f.

Zacconi, L., 46.

Zarlino, G., 6, 9, 11, 25, 27, 32, 35, 37, 42,
46, 50 f, 59, 113, 118, 121, 144, 147, 164,
194 f.

Zeisung, H., 142.

Zipoli, D., 191.

Intervals with i

Juperpartici

ilar Ratios

Ratios

Intervals

Cents

Page References in Text

2:1

octave

1200

passim

3:2

perfect 5th

702

passim

4:3

perfect 4th

498

passim

5:4

major 3rd

386

passim

6:5

minor 3rd

316

passim

7:6

minor 3rd

267

18, 19 (Table 13), 22 f, 30 f.

8:7

maximum tone

234

19 (Table 13), 20, 23 f, 152.

9:8

major tone

204

passim

10:9

minor tone

182

passim

11:10

minimum tone

165

21 f, 154.

12:11

semitone

150

18, 21, 152-154.

13:12

•

139

23, 154.

14:13
15:14

approximation to

meantone diatonic

semitone

128
119

23.

17 (Table 5), 18, 23, 30 f, 152-1E

16:15

just diatonic
semitone

112

passim

17:16
18:17

semitone

approximation to

semitone of equal

temperament

105
99

57, 141, 153 f.

8, 57-64, 141, 153 f, 186.

19:18

semitone

17 (Table 7), 18, 57, 141, 154.

20:19

•

16 (Table 2), 17 (Table 7),

19 (Tables 13 and 14), 18, 141, 1

21:20

20, 22, 153 f.

22:21

•

18, 23, 151, 153.

227

TUNING AND TEMPERAMENT

Ratios

Intervals
approximation to

Cents

Page References in Text